WAVE GENERATION AND SHAPING
*
McGRAWHILL ELECTRICAL AND ELECTRONIC
ENGINEERING SERIES
Frederick Emmons Terman, Consulting Editor
W. W. Harman, Hubert Heffner, and J. G. Truxal, Associate Consulting Editors
AHRENDT AND SAVANT Servomechanism Practice
ANGELO Electronic Circuits
ASELTINE Transform Method in Linear System Analysis
ATWATER Introduction to Microwave Theory
BAILEY AND GAULT Alternatingcurrent Machinery
BERANEK Acoustics
BRENNER AND JAVID Analysis of Electric Circuits
BROWN Analysis of Linear Timeinvariant Systems
BRUNS AND SAUNDERS Analysis of Feedback Control Systems
CAGE Theory and Application of Industrial Electronics
CAUER Synthesis of Linear Communication Networks
CHEN The Analysis of Linear Systems
CHEN Linear Network Design and Synthesis
CHIRLIAN AND ZEMANIAN Electronics
CLEMENT AND JOHNSON Electrical Engineering Science
COTE AND OAKES Linear Vacuumtube and Transistor Circuits
CUCCIA Harmonics, Sidebands, and Transients in Communication Engineering
CUNNINGHAM Introduction to Nonlinear Analysis
EASTMAN Fundamentals of Vacuum Tubes
EVANS Controlsystem Dynamics
FEINSTEIN Foundations of Information Theory
FITZGERALD AND HIGGINBOTHAM Basic Electrical Engineering
FITZGERALD AND KINGSLEY Electric Machinery
FRANK Electrical Measurement Analysis
FRIEDLAND, WING, AND ASH Principles of Linear Networks
GEPPERT Basic Electron Tubes
GHOSE Microwave Circuit Theory and Analysis
GREINER Semiconductor Devices and Applications
HAMMOND Electrical Engineering
HANCOCK An Introduction to the Principles of Communication Theory
HAPPELL AND HESSELBERTH Engineering Electronics
HARMAN Fundamentals of Electronic Motion
HARMAN Principles of the Statistical Theory of Communication
HARMAN AND LYTLE Electrical and Mechanical Networks
HARRINGTON Introduction to Electromagnetic Engineering
HARRINGTON Timeharmonic Electromagnetic Fields
HAYT Engineering Electromagnetics
HILL Electronics in Engineering
HUELSMAN Circuits, Matrices, and Linear Vector Spaces
JAVID AND BRENNER Analysis, Transmission, and Filtering of Signals
JAVID AND BROWN Field Analysis and Electromagnetics
JOHNSON Transmission Lines and Networks
KOENIG AND BLACKWELL Electromechanical System Theory
KRAUS Antennas
KRAUS Electromagnetics
KUH AND PEDERSON Principles of Circuit Synthesis
LEDLEY Digital Computer and Control Engineering
LEPAGE Analysis of Alternatingcurrent Circuits
LEPAGE AND SEELY General Network Analysis
Jf J,V ¥ L tY T ?; t AND REHBER G Linear Circuit Analysis
^Im^^ GIBB0NS Transis t°rs ^d Active Circuits
™™ t™ TRUXAL Introductory System Analysis
^v^S A ™ ™UXAL Principles of Electronic Instrumentation
MIU £ AN VD V TRUXAL , SigM,S aDd SyStemS iD E,ectrical Engineering
^I™ A ^ . y acuura  tub « and Semiconductor Electronics
MILLMAN AND SEELY Electronics
MILLMAN AND TAUB Pulse and Digital Circuits
MISHKIN AND BRAUN Adaptive Control Systems
MOORE Travelingwave Engineering
NANAVATI An Introduction to Semiconductor Electronics
„tZZII . Electronic Switching, Timing, and Pulse Circuits
PFEIfJf* 1 McW Q H ORTER Electronic Amplifier Circuits
ytt.lt thti Linear Systems Analysis
REZA An Introduction to Information Theory
REZA AND SEELY Modern Network Analysis
ROGERS Introduction to Electric Fields
RYDER Engineering Electronics
o£^ AR J Z Information Transmission, Modulation, and Noise
bh.&L,Y Electromechanical Energy Conversion
SEELY Electrontube Circuits
SEELY Electronic Engineering
SEELY Introduction to Electromagnetic Fields
SEELY Radio Electronics
SEIFERT AND STEEG Control Systems Engineering
SIEGMAN Microwave Solid State Masers
SISKIND Directcurrent Machinery
SKILLING Electric Transmission Lines
SKILLING Transient Electric Currents
SPANGENBERG Fundamentals of Electron Devices
SPANGENBERG Vacuum Tubes
STEVENSON Elements of Power System Analysis
STEWART Fundamentals of Signal Theory
STORER Passive Network Synthesis
STRAUSS Wave Generation and Shaping
TERMAN Electronic and Radio Engineering
TERMAN AND PETTIT Electronic Measurements
THALER Elements of Servomechanism Theory
™ *™ B *°™ Ana,ySiS 8Dd DesigD ° f Feedback Contr °> S * st <>n*
THALER AND PASTEL Analysis and Design of Nonlinear Feedback Control
Systems
THOMPSON Alternatingcurrent and Transient Circuit Analysis
TOU Digital and Sampleddata Control Systems
TRUXAL Automatic Feedback Control System Synthesis
VALDES The Physical Theory of Transistors
VAN BLADEL Electromagnetic Fields
WEINBERG Network Analysis and Synthesis
WILLIAMS AND YOUNG Electrical Engineering Problems
BROOKLYN POLYTECHNIC INSTITUTE SERIES
angelo ■ Electronic Circuits
lynch and truxal • Signals and Systems in Electrical Engineering
Combining Introductory System Analysis
Principles of Electronic Instrumentation
mishkin and braun ■ Adaptive Control Systems
Schwartz • Information Transmission, Modulation, and Noise
stbauss ■ Wave Generation and Shaping
Wave Generation and Shaping
LEONARD STRAUSS
ASSOCIATE PROFESSOR OF ELECTRICAL ENGINEERING
POLYTECHNIC INSTITUTE OF BROOKLYN
INTERNATIONAL STUDENT EDITION
McGRAWHILL BOOK COMPANY, INC.
New York Toronto London
KOGAKUSHA COMPANY, LTD.
Tokyo
WAVE GENERATION AND SHAPING
INTERNATIONAL STUDENT EDITION
Exclusive rights by Kogakusha Co., Ltd. for manufacture and export
from Japan. This book cannot be reexported from the country to
which it is consigned by Kogakusha Co., Ltd. or by McGrawHill Book
Company, Inc. or any of its subsidiaries.
I
Copyright © 1960 by the McGrawHill Book Company, Inc. All
rights reserved. This book, or parts thereof, may not be reproduced
in any form without permission of the publishers. Library of Con
gress Catalog Card Number 609861
TOSHO PRINTING CO., LTD., TOKYO. JAPAN
TO ROSEMARY
PREFACE
Time and time again during the last fifty years the electrical engineer
has been called upon to generate a signal having some specified geometry.
Either through intuition and experimentation or on the basis of a detailed
analysis of the specific problem, the first crude circuits were designed,
improved, and modified. With the advent of the transistor, not only
were many of the vacuumtube circuits adapted, but new ones, making
use of the special transistor characteristics, were devised. Thus today
hundreds of oscillators, multivibrators, and linear sweeps are described
in the literature; usually each one is treated as a separate entity.
When only crude methods of analysis were at the disposal of the
engineer and when the number of useful active circuits was small, it was
necessary for the texts to discuss them all in detail. But with the current
state of the art, an encyclopedic study becomes prohibitive, if, in fact,
such a work would even be desirable.
The objective of this text is to present a logical, unified approach to
the analysis of those circuits where the nonlinearity of the tube or
transistor is significant. A developmental treatment is followed through
out as we focus on the essential features of practical wavegenerating
and shaping circuits. To this end, the text is arbitrarily divided into
five sections: models and shaping, timing, switching, memory, and
oscillations. It is, of course, impossible not to step across these bounds
in discussing specific circuits and examples have been chosen so that the
basic ideas will arise naturally from the discussion. In most cases the
analysis is sufficiently detailed so that the techniques may be applied to
other existent, and not as yet existent, circuits. Transistors and vacuum
tubes are used almost interchangeably to support the contention that the
basic mode of operation is independent of the active element employed.
The organization, which is described in some detail below, is a result
of experimentation with class notes over several years. Because this
work is primarily a text, designed for a senior or graduate course of one
or two semesters, it is assumed that the reader is familiar with the tran
sient analysis of linear networks and with simple vacuumtube and
transistor amplifiers. However, since the viewpoint presented may
be somewhat different from that previously encountered, some review
X PREFACE
material is incorporated in the body of the text at the point where it is
first needed.
Part 1. Models and Shaping. Since we are interested in the transient
response of the piecewiselinear models to various excitations, the first
chapter presents approximation methods which can be employed for
the rapid solution of linear circuits.
In Chapter 2 a grossly nonlinear element, the diode, is introduced.
Once its voltampere characteristic is approximated by an ordered series
of line segments, linear algebraic equations may be written to define the
behavior in each region. Most of the problems met in more complex
circuits make their initial appearance when the diode is combined with an
energystorage element. Combinations of diodes and resistors can be
used to represent, in a piecewise manner, almost any arbitrary character
istic. For solution, the complex circuit is reduced to a sequence of linear
networks and the complex problem reduces to finding the boundaries
between the various regions (breakpoint analysis). Chapter 3 reinforces
the concepts of Chapter 2 with additional examples of multiplediode
networks.
The piecewiselinear model is extended to multiterminal devices (the
tube and transistor) in Chapter 4, relying on the conduction or back
biasing of ideal diodes to differentiate the regions of operation. In
order to represent the inherent amplification of the active element, the
model includes one or more controlled sources.
Part 2. Timing. The next four chapters deal with the linear sweep.
Since the essential portion of any timed circuit is the exponential charging
of an energystorage element, the problem of analysis resolves itself
into finding the timeconstant, the initial, the steadystate, and the final
sweep voltage.
In the simple voltage sweep of Chapter 5, a gas tube is used as the
switching element. This choice makes it possible to treat synchroniza
tion without having recourse to the more complex circuit configurations
of the later chapters.
Linearization of the sweep always involves some form of feedback.
The aim is to approximate constantcurrent charging of the sweep capaci
tor. Vacuumtube circuits are analyzed in Chapter 6, but a discussion
of the transistor equivalents is deferred until Chapter 7. In the Miller
sweep of Chapter 6, the calculation of the initial jumps, overshoots, and
recovery exponentials, from the individual circuit models, lays the
groundwork for the later analysis of the multivibrator and the blocking
oscillator. Furthermore, the separate consideration of the recovery time
and the switching problem leads to the construction of a sweep system, the
phantastron. Chapter 8 applies the same techniques of analysis to the
current sweep.
PREFACE XI
Part 3. Switching. When a closedloop system contains active ele
ments and when the loop gain is positive and greater than unity, we
call it a switching circuit. If any timing networks are included in the
transmission path, the circuit exhibits quasistable behavior. The same
effect can be obtained by shunting a negativeresistance device with an
energystorage element.
Chapters 9 and 10 discuss a closedloop regenerative switching circuit,
the multivibrator. We are mainly concerned with where and how to
begin the analysis. A guess serves as the convenient starting point, and
the circuit calculations are checked to see if they yield consistent results.
If a contradiction arises, it simply indicates that the first guess was wrong.
In the multivibrator both tubes and transistors are active only during the
switching interval. At all other times one is cut off or the other is
saturated. Hence the exponential timing, which sustains the limiting,
occurs within the cutoff zone, and recovery usually depends on the
saturated tube or transistor. After the appropriate models are drawn,
the calculations involve finding the initial and switching points of the
timing network.
Many devices exist whose drivingpoint voltampere characteristic
exhibits a negativeresistance region. The most useful treatment of
switching circuits containing these elements is from the viewpoint
expressed in Chapter 11. Switching and timing are first treated with the
aid of a postulated ideal device. Even though the waveshapes and
trajectories obtained are approximately correct, the use of an ideal
negative resistance leads directly to a stability criterion that is incon
sistent with the physical device. Since the reasons that the contradictory
results arose are pointed out later, this chapter also illustrates the dangers
inherent in overidealizing a model.
The ideas presented in Chapter 11 may well be considered as unifying
concepts. These are presented rather late in the text so that the student
may better appreciate the limitations as well as the advantages of this
viewpoint. The negativeresistance treatment is useful for understand
ing the operation of all sweeps and switching circuits even where it is
difficult to isolate the two terminals across which the negative resistance
is developed. It is, however, a convenient design method only for those
circuits which employ such special devices as the unijunction and pnpn
transistors and the tunnel diode.
Not all circuits can, in fact, be solved completely. But the methods
developed in the earlier sections of the text, when applied to one which
cannot be, for example, the blocking oscillator of Chapter 12, do yield a
considerable understanding of how the circuit behaves.
Part 4. Memory. Since a hysteresis loop indicates memory, magnetic
and dielectric materials are ideally suited for information storage.
xii PREFACE
Chapter 13 examines briefly the terminal characteristics of a core con
structed out of ideal squareloop material. Furthermore, a clearly
defined time is needed to saturate the core, and although not representable
in a piecewiselinear manner, the drivingpoint impedance exhibits two
distinct regions. This enables us to use either a core or a ferroelectric
device as the timing element in a switching circuit.
Part 5. Oscillations. Although most of the circuits previously dis
cussed depended for their timing on a single energystorage element, at
least two are necessary in order for the system to have a pair of complex
conjugate poles located in the right half plane.
The sinusoidal oscillator is first treated as an almost linear feedback
system (Chapter 14) which allows the separation of the frequency,
amplitude, and gaindetermining element. The role of each oscillator
essential is individually examined, with emphasis on minimizing the
distortion and maximizing the stability. Since the signal produced is
almost sinusoidal, the amplitude is readily found by plotting a system
describing function.
Chapter 15 returns to the negativeresistance viewpoint and describes
the solution of a specific nonlinear differential equation. Here topological
constructions, which supplement analytic methods, yield the waveshape
and amplitude of oscillations. Hence the text ends by introducing a new
topic and not by saying the last word on an old one.
While preparing the manuscript, I received a great deal of assistance
and encouragement from many people. I should like to acknowledge the
contributions made by those colleagues who taught sections of the course
for which this text was written. In stimulating technical discussions, I
received the benefit of their ideas and thinking on many topics, much of
which caused me to modify the original treatment. They also corrected
errors which inadvertently crept into the original class notes, and they
prepared some of the problems. The secretarial and drafting assistance
made available by the Polytechnic Institute of Brooklyn is greatly
appreciated. Finally, I wish to thank all members of the Electrical
Engineering Department of the Institute for their generous encourage
ment during the project. The friendly and cooperative atmosphere made
the writing of this book almost a pleasure.
Leonard Strauss
CONTENTS
Preface ix
PART 1— MODELS AND SHAPING
Chapter 1. Linear Wave Shaping 3
11. Introduction 3
12. Initial Conditions 3
13. Solution of Periodically Excited Circuits 8
14. Differentiation 10
15. Integration 12
16. Summation 13
17. Approximate Solutions of the Singkrenergy Case 14
18. Doubleenergystorageelement Systems 16
19. Compensated Attenuators 19
Chapter 2. Diode Waveshaping Techniques 26
21. Passivenonlinearcircuit Representation 26
22. Physical Diode Model 32
23. Voltage Clipping Circuits 33
24. Single Diode and Associated Energystorage Element — Some General
Remarks 37
25. Diode Voltage Clamps 38
20. Current Clippers 45
27. Current Clamps 46
28. Arbitrary Transfer and VoltAmpere Characteristics 48
29. Dead Zone and Hysteresis 55
210. Summary 59
Chapter 3. Diode Gates 67
31. Application 67
32. or Gate 67
33. and Gate 72
34. Controlled Gates 75
35. Diode Arrays 78
36. Diode Bridge — Steadystate Response 79
37. Diode Bridge — Transient Response 85
38. Concluding Remarks on Diode Gates 88
Chapter i. Simple Triode, Transistor, and Pentode Models and Circuits . 96
41. Triode Models 96
42. Triode Clipping and Clamping Circuits 107
43. Triode Gates . 110
44. Transistor Models 116
45. Simple Transistor Circuits 125
xiii
XIV CONTENTS
46. Transistor Gates 130
47. Pentodes 133
48. Summary 135
PART 2— TIMING
Chapter 6. Simple Voltage Sweeps, Linearity, and Synchronization 147
61. Basic Voltage Sweep 147
52. Gastube Sweep 148
53. Thyratron Sweep Circuits 152
54. Sweep Linearity 154
55. Synchronization 157
56. Regions of Synchronization 161
Chapter 6. Vacuumtube Voltage Sweeps 172
61. Introduction 172
62. Linearity Improvement through Current Feedback 172
63. Bootstrap Sweep 175
64. Miller Sweep 178
65. Recoverytime Improvement . 186
66. Sweep Switching Problems 189
67. Pentode Miller Sweep 190
68. The Phantastron 194
69. Miller Sweep and Phantastron — Screen and Controlgrid Voltage Calcu
lations 198
Chapter 7. Linear Transistor Voltage Sweeps 206
71. Constantchargingcurrent Voltage Sweep 206
72. Bootstrap Voltage Sweep 209
73. Miller Sweep 214
74. Compound Transistors 216
Chapter 8. Linear Current Sweeps 222
81. Basic Current Sweeps 222
82. Switched Current Sweeps 226
83. Currentsweep Linearization 232
84. A Transistor Bootstrap Sweep 235
85. Constantcurrentsource Current Sweep 238
PART 3— SWITCHING
Chapter 9. PlateGrid and CollectorBasecoupled Multivibrators 247
91. Basic Multivibrator Considerations 247
92. Vacuumtube Bistable Multivibrator 249
93. Transistor Bistable Multivibrator 255
94. A Monostable Transistor Multivibrator 258
95. A Vacuumtube Monostable Multivibrator 261
96. Vacuumtube Astable Multivibrator 260
97. Transistor Astable Multivibrator , . . . . 269
98. Inductively Timed Multivibrators 274
99. Multivibrator Transition Time 277
910. Multivibrator Triggering and Synchronization 280
CONTENTS XV
Chapter 10. Emitter coupled and Cathodecoupled Multivibrators .... 289
101. Transistor Emittercoupled Multivibrator — Monostable Operation . . 289
102. Modes of Operation of the Emittercoupled Multivibrator .... 293
103. Monostable Pulse Variation 301
104. Emittercoupled Astable Multivibrator 303
105. Cathodecoupled Monostable Multivibrator 307
106. Limitation of Analysis 310
Chapter 11. Negativeresistance Switching Circuits 315
111. Basic Circuit Considerations 315
112. Basic Switching Circuits 317
113. Calculation of Waveshapes 321
114. Voltagecontrolled NLR Switching Circuits 325
115. NLR Characteristics — Collectortobasecoupled Monostable Multi
vibrator 326
116. Some Devices Possessing Currentcontrolled NLR Characteristics 329
117. Some Devices Possessing Voltagecontrolled Nonlinear Characteristics. 336
118. Frequency Dependence of the Devices Exhibiting NLR Characteristics 341
119. Improvement in Switching Time through the Use of a Nonlinear Load . 346
1110. Negativeimpedance Converters 349
Chapter 12. The Blocking Oscillator 357
121. Some Introductory Remarks 357
122. An Inductively Timed Blocking Oscillator 359
123. Transformer Core Properties and Saturation ....:... 365
124. Capacitively Timed Blocking Oscillator 367
125. Astable Operation 370
126. A Vacuumtube Blocking Oscillator 372
127. Some Concluding Remarks 380
PART 4— MEMORY
Chapter 13. Magnetic and Dielectric Devices as Memory and Switching Ele
ments 387
131. Hysteresis — Characteristics of Memory 387
132. Ferromagnetic Properties 389
133. Terminal Response of Cores 393
134. Magnetic Counters 400
135. Coretransistor Counters and Registers 406
136. Magnetic Memory Arrays 408
137. Coretransistor Multivibrator 410
138. Properties of Ferroelectric Materials 414
139. Ferroelectric Terminal Characteristics 417
1310. The Ferroelectric Counter 419
1311. Coincident Memory Arrays 421
PART 5— OSCILLATIONS
Chapter 14. Almost Sinusoidal Oscillations — The Linear Approximation . 431
141. Basic Feedback Oscillators 431
142. Characteristics of Some RC and LR Frequencydetermining Networks . 435
143. Transistor Feedback Oscillators 442
XVi CONTENTS
144. Tunedcircuit Oscillators 445
145. Frequency Stabilization 455
146. Amplitude of Oscillations 460
147. Amplitude Stability 471
Chapter 15. Negativeresistance Oscillators 477
151. Basic Circuit Considerations 477
152. Firstorder Solution for Frequency and Amplitude 482
153. Frequency of Oscillation to a Second Approximation 487
154. Introduction to Topological Methods 492
155. The Lienard Diagram 499
156. Summary 505
Name Index 511
Subject Index 513
PART 1
MODELS AND SHAPING
CHAPTER 1
LINEAR WAVE SHAPING
11. Introduction. Often the control and instrumentation engineer
finds himself faced with the problem of producing a designated complex
waveform. Usually his simplest approach is to divide the problem into
two parts: first, the generation of simple waveshapes and, second, some
type of operation on them to achieve the desired final result. If the
required waveform is of a sufficiently complex nature, several substages
of shaping may be needed, with combination taking place in the final
stage. Generation always involves the use of active elements, and
although they also perform a function, shaping is predominantly effected
by the circuit's passive components. One might even go so far as to say
that a major portion of the process of wave generation itself consists in
passive element shaping.
In this chapter we shall examine the behavior of linear passive circuits
with a view to solving for their time response to various input wave
shapes. Exact analysis is, of course, possible, but often so tedious that
the competent engineer looks instead for reasonable approximations.
Our aim, therefore, is to set up conditions whereby solutions may be
obtained rapidly, almost by inspection, avoiding involved algebraic
manipulation. This, in fact, becomes possible only under very specific
circumstances, which must be clearly delineated. Particular emphasis
will be placed on the treatment of circuits containing only a singleenergy
storage element, not only because these appear quite frequently, but also
because it is here that the approximation techniques are most fruitful.
Sometimes even multipleenergystorage circuits may be reduced to a
number of separate singleenergy circuits, which are then individually
solved. The solutions are finally combined, yielding the approximate
total response of the original complex circuit.
In our effort to develop facility in utilizing simplifying approximations,
i.e., how to make them and when to make them, we shall start by con
sidering some very basic circuits. Since we already know methods of
finding their exact response to simple inputs, we can concentrate atten
tion on the methodology of the solution.
12. Initial Conditions. We shall first consider the transient response
of a passive linear network, i.e., the response to some abrupt change in
3
4 MODELS AND SHAPING [CHAP. 1
either the external excitation or the internal circuit. We know that
the solution will be directly related to the poles of the system transfer
function, and for poles at pi, p 2 , and p 3) the response to a step will always
have the form
r(t) = A + Aie>»' + A i e'" t + A z e">'
One exponential term is due to each energystorage element included
in the network. This type of response is readily found by the classical
method of writing and solving the system's integrodifferential equations
or by modern transformation analysis.
If, however, we restrict our interest to relatively simple systems, i.e.,
those containing one, or sometimes two, widely separated poles, then
these may often be found by direct inspection of the circuit. They will
correspond to the negative reciprocal of the time constants.
20v:=:
Fig. 11. Singleenergystorageelement circuit.
In a circuit containing only a singleenergystorage element, such as
the one shown in Fig. 11, the response to a unit step will be given by
e (t) = A + Be"'
(1D
The first parameter of interest, the time constant, is simply the product
of C and the total resistance seen by the capacitor. When the switch
is in position 2,
t 2 = (fli + R.)C p 2 =  r (12)
T 2
Upon switching to position 3, both the time constant and pole location
change; they now become
r 3 = RxC V z =   (13)
The time response is of the form given in Eq. (11) ; coefficients, of course,
will differ.
Since a prerequisite to the complete solution is the value of the various
coefficients, a slight digression as to the means of their evaluation is in
order before proceeding to the general solution of this circuit. But
these coefficients depend upon the initial conditions; therefore the first
Sec. 12] lineab wave shaping 5
question to ask is, What determines these conditions? Obviously, one
part of the answer is the known external constraints imposed on the
circuit, i.e., the excitation function and the various circuit changes. The
second half of the answer is the internal constraints determined by the
energystorage elements present within the circuit, in this case a single
capacitor.
The voltage drop across any capacitor is related to circuit current flow
as expressed in the differential equation
e c =±fidt = Q (14)
This equation says that, in a physical system, the accumulation of charge
takes finite time unless an infinite current flows. Consequently the
terminal voltage cannot change instantaneously. In an ideal system
such instantaneous changes may be forced by placing a short circuit or
an ideal battery across the capacitor terminals or by injecting an infinite
pulse of current. These do not exist in practice, but sometimes the
charge or discharge time is so very fast compared with the total time
range of interest that the small time interval involved may be ignored,
and we can say that the capacitor voltage has jumped to its final value.
From the above discussion, the first circuit constraint can now be stated :
the voltage across the capacitor must be continuous across regions of circuit
or excitation change. Carrying this argument a step further leads us
to the additional conclusion that all sudden changes in circuit voltage
will be distributed across the various noncapacitive portions of the
circuit — across the resistors and inductances. The manner of division
depends on the particular circuit configuration under investigation.
One can derive a second condition from Eq. (14) by noting that the
voltage across C continues to change as long as any current flows. The
final steadystate voltage is reached when i c = 0. In a simple network
it will always be the dc Thevenin equivalent voltage appearing across
the capacitor.
We conclude that the constant A in Eq. (11) represents the steady
state response and may be evaluated by considering the behavior at
t = » . The coefficient B is related to the behavior at t = and would
be found from the initial conditions.
As an example, consider the time response of the circuit of Fig. 11.
Its behavior is the following: at t = the switch is thrown from position 1
to 2; once the voltage across the capacitor reaches the predetermined
final value, Ef = 15 volts, the switch is moved to position 3, remaining
there.
Rather than attempt to solve the entire problem in a single step, it is
advisable to divide it into three simple parts and to solve each in proper
6
MODELS AND SHAPING
[Chap. I
time sequence. The complete solution will then be the aggregate of the
individual answers. The three parts are:
1. Behavior at t = 
2. Response from I = 0+ to the time t t at which e„(t/) = E t
3. Response from t = t/ to »
Note that the change of state determines the problem division, with
each part holding for a specific circuit configuration. In proceeding, the
solution of each part will start from its own zero time as if it were a com
pletely independent problem. Numerical subscripts corresponding to
the particular part indicate the time region under examination.
1. Solution for t < Oi:
e i = — 10 volts
2. System response for (h < t < t/2. In this region the solution is
given by Eq. (11). We are now ready to evaluate the coefficients from
the specific initial conditions holding.
20
15
**
*
t
1 \r 3
*o(0
V
/
*/2
1.79r 2
t
E a
10
Fig. 12. Total time response of the circuit of Fig. 11.
Because the voltage of C must be continuous across the switching
interval (circuit change), the initial value in this region (En) will be the
same as the final value of part 1 (E/i). At t = + ,
e„ 2 (0+) = E it = E f i = 10 volts = A 2 + B 2
(15)
Furthermore, the output now charges toward a steadystate value E,,i,
found at t = »
e<, 2 (°°) = E„i = 20 volts = A 2 (16)
Solving Eqs. (15) and (16) and substituting into (11) yields
e 02 (<) = E„2  (E.. t  E i2 )er<i"
= 20  30e« r > (17)
The final value of part 2 is found by substituting t fi into Eq. (17).
Sec. 12] linear wave shaping 7
e i(t/ t ) = E f i = 15 volts = E..2  (E„ t  E it )e'»i r *
Solving for t/i yields
, , E„2 — En , 20 — ( — 10) t _ n ,, .
tfi = Ti ln E„ t  E n = T2 ln 2015 = L79t2 (1 ' 8)
3. System response for < /2 < t < °o or 3 < t < °o. By the same
methods used for part 2, the new defining equation becomes
(19)
e„ 3 (0 = E.,3
 (E.. %  E i3 )e""
The
t initial and
steadystate voltages in
region 3 are
(a)
t = 3
En =
Efi = 15
(b)
t = 00
e s =
E.,3 =
Substituting these conditions into Eq. (19) results in the final decay
equation
e oS (0 = E/tfT'i" = 15e""
Figure 12 is the plot of the total solution, with the response of each
portion beginning immediately after that of the previous part.
In the solution for part 2, the circuit responded as if no change were
scheduled. It could not predict the future, and therefore, in solving
the problem, we must be careful lest our foreknowledge lead to fallacious
reasoning, i.e., the incorrect substitution of E/t instead of E„ 2 for the
steadystate value. Upon entering region 2 the response is toward the
final value calculated as if the circuit were invariant.
When the circuit contains an ideal inductance, additional constraints
appear.
•Lig (MO)
From Eq. (110) it may be seen that an infinite voltage is required to
change the current instantaneously. As this is an impossible situation
in any physical circuit, our first conclusion is that the current through
the coil must be continuous across circuit or excitation changes. A second
constraint, also derivable from the above equation, is that steady state
is finally reached when there is no further time rate of change of current,
i.e., when ex, = 0. These conditions are just the duals of the ones found
for the capacitor. The overlapping constraints of the two energy
storage elements forbid any instantaneous changes of current, and of
capacitor or resistor voltage, in a series RLC circuit. Because of the
physical nature of the various circuit elements, they will all have associ
ated stray capacity and lead inductance. Therefore all circuits contain
8
MODELS AND SHAPING
[Chap. 1
more than one type of energystorage element. These parasitic elements
act to slow down the rapid changes found when we consider a simple
ideal BL or BC circuit. But if the time range of interest is sufficiently
long, the secondorder effects may be neglected.
13. Solution of Periodically Excited Circuits. After the sudden
application of a periodic input to a passive network, the output itself
will eventually become periodic in nature. The transient dies out in
four time constants, leaving the steadystate response. The time
required for this, in terms of the number of cycles of the input wave
shape, depends upon the relative values of the time constant and the
period of the excitation function. If the time constant is much smaller
than the input period, the periodic output appears within one or at
most a few cycles and it can be found by following the behavior cycle by
cycle.
R
9 WV
u
ei(t)
*6
Region I
Region U
/
*r,—
T z
t
\
erf)
e 2 (t)
Fig. 13. Circuit for the study of recurrent boundary conditions.
On the other hand, when the ratio of the time constant to the period
of the input is relatively large, periodicity will not be reached until an
extremely large number of cycles have passed. During the buildup,
the capacitor charges to the average level of the voltage appearing across
its terminals and the direct current through a coil increases from zero
to the average circuit value. To attempt to start at zero time and follow
the response cycle by cycle until final periodicity is a ridiculous approach
to the solution of this class of problems — it will only lead to frustration.
A more judicious treatment would be the application of the results
of the previous section, with the addition of the known periodicity of
the solution. This enables us to write recurrent initial conditions; i.e.,
the starting point of any one cycle is the same as that of the next cycle.
As an example, we shall now examine the RL circuit of Fig. 13. The
rectangular input shown has been applied for a sufficiently long time
so that the output has already reached the final periodic form shown in
Fig. 14.
The solution during either portion of the input signal is
e t (t) = A + Ber'ir
L
R
Sec. 13]
LINEAR WAVE SHAPING
9
Except for the time constant, this equation is of the same form as the
one found for the RC circuit [Eq. (11)]. Since the inductance prevents
abrupt changes in the current, the resistor voltage will be invariant
across the discontinuities of the input and any abrupt jump in the driving
voltage will be reflected across the coil. We can show this by considering
the voltage drops around the loop:
ci(0 = e B + e L = iR + e£ (111)
Across the boundary of the two regions the input voltage jumps, and
by considering this change as Aei(i), we see that
Ae x (t) = MR + Ae L
But since the inductance constrains At to be zero, the complete jump
in the input voltage must immediately appear across the inductance.
Fig. 14. Output voltage across L in the circuit of Fig. 13.
Starting the solution at the beginning of region I, which holds for
Oi < t < T\, the first set of initial conditions is
E.,i = En = unknown
Therefore the solution in this region is
««(0 = Ear*" (112)
and the final value, occurring at t = T\, may be expressed as
E n = «■ (TO = Bar*" (113)
Upon entering region II, the input voltage drops from E a to E b , with
the same change immediately appearing across the inductance. The final
output of region I was E/i. The initial value of region II must be
Em = E/i — (E a — Ei,)
10 MODELS AND SHAPING [CHAP. 1
In region II, holding for 2 < t < Ti, E„ n = 0. The equation for the
time response can be written
«m(0 = [E n ~ (E.  E h )]e'» (114)
At t = T h
E m = e ai (T 2 ) = [E n  (E a  E b )]e™ (115)
But the initial voltage of region I, and all other voltages which have
been written in terms of it, are still unknown. We must now apply the
known conditions of periodicity in order to find the final answer. At the
end of region II the input voltage jumps from E\, to E a and the output
voltage changes by the same amount. Periodicity tells us that the new
value upon reentering region I is the original assumed E a .
E a = E m + (E.  E h ) (116)
Substituting Eq. (116) into (113) results in
E n = [E m + {E a  E b )]e^ir (117)
Equations (115) and (117) each involve only the two unknowns En and
Efii', all other terms are constants of the input waveshape or the circuit.
The simultaneous solution of the above equations completes the analysis.
If the output of interest were taken across the resistor instead of the
inductance, the technique of solution would still be the same, the prin
cipal difference being in the initial conditions over each portion of the
cycle. Current is continuous, and therefore resistor voltage must also
be continuous in the RL circuit. RC series circuits are treated in a
similar manner, applying, however,
? l(  ? their special boundary conditions; i.e.,
I < I all voltage jumps appear across the
* l l R S * 2  circuit resistance.
1  i Even when the inputs are nonrec
Fig. 15. Differentiator. tangular, the shape and equation of
their response over any period or
portion thereof usually can be obtained by many convenient methods.
The known periodicity supplies the additional information necessary for
a complete solution.
14. Differentiation. A series RC circuit (Fig. 15) will, contingent
on the satisfaction of certain conditions, have an output approximately
proportional to the derivative of the input. These conditions may best
be determined by examination of the circuit differential equation
e,(t) = Ri = ei(t)  ^ I i dt (118)
Sec. 14] linear wave shaping 11
Since the voltage across a resistor is directly proportional to the current
flow through it, for the output to be proportional to the derivative, the
current must also have this relationship. Satisfaction of this necessary
condition is made possible only by maintaining the output voltage small
with respect to the input. Then from Eq. (118),
Therefore
and
ei(0 £g j idt
, dei(t)
i^C
csSRC
dt
deijt)
dt
(119)
The major portion of the circuit voltage drop is developed across C
only when the time constant is small compared with the time range of
interest. At discontinuities the total change of input voltage appears
at the output and the circuit only roughly approximates a differentiator.
19.6
•2ft)
J
19.6
i
V T200^«sec
r
Fig. 16. Differentiator input and output waveshapes.
With a sufficiently small time constant, the exact output waveshape is
reasonably close, except at the discontinuities, to the one found by assum
ing a perfect differentiator. The exact derivative at a discontinuity
of the input waveshape is infinity. Since instantaneous jumps do not
exist in nature, the full change in the input voltage, which appears across
the output at the input "discontinuities," is so very much larger than the
normal small output signal that the circuit may well be said to approxi
mate the derivative even here. Thus, after checking the time constant
against the input waveshape, the output can be drawn by inspection and
the voltage values calculated later. Figure 16 illustrates this process
for an RC circuit having a time constant of 200 /usee.
12 MODELS AND SHAPING [CHAP. 1
Because the capacitor will not pass direct current, the periodic response
to periodic inputs must have zero average value. The capacitor will
store a charge proportional to the average level of the input signal, and
its voltage will fluctuate almost between the input extremes.
An RL series circuit having a small time constant also differentiates.
Here the output must be taken across the inductance because the basis
of differentiation in this circuit is the proportionality of the inductive
voltage to the time rate of the circuit current change. By maintaining
H almost the full input voltage drop across
the series resistor, we ensure the cur
t
* rent's dependency only on this resist
ed C^ e 2 (t) ance and guarantee good differentiation.
I 15. Integration. Provided that the
° time constant is very long, the capacitor
Fig. 17. Integrator. voltage in a series RC circuit is pro
portional to the integral of the input
voltage (Fig. 17). Again the required condition may be found from
an examination of the circuit differential equation
hi
eiif) = g idt = e x (0  'Ri (120)
When the output is small compared with the input, the approximate
relationship
ei(<) ^ Ri
results, and therefore
e,(.t)^4n ( e > dt (121)
If the output voltage rises above a relatively small value, the current,
* = (ei — e 2 )/R, becomes dependent on e% and the circuit will no longer
integrate satisfactorily. An excessively rapid rise in the output voltage
within any period is directly dependent on the rate of charging of C.
A slow charging rate remains possible only through keeping the time
constant long compared with the time over which the integral is desired. C
eventually charges to the dc level of the input with the circuit time con
stant, and the output voltage shifts accordingly. For a signal impressed
at t = 0, Fig. 18 shows both the input and its integrated output. Even
tually the output will shift down until it has a zero average value.
The output voltage, at any time, is proportional to the area under the
curve of the input signal from zero to that point. Consequently its
shape may be sketched directly from the curve and the coordinates found
from Eq. (121). In order to calculate the voltage change at the output
Sec. 16]
LINEAH WAVE SHAPING
13
over any time interval, the input volttime area is simply divided by the
time constant.
A largetimeconstant RL circuit also integrates with the small output
voltage now appearing across R. Physical inductances always have
associated winding resistance and, for large coils, appreciable stray
R 2
VWW 9
Fig. 18. Input signal and integrated output. Input applied at J = 0.
capacity. The impossibility of obtaining an ideal element usually pre
cludes the use of RL circuits for many waveshaping functions. On the
other hand, almost ideal capacitors are readily available. Using the new
plastic dielectrics, leakage resistance, the major parasitic element, is of
the order of 10 5 megohms, a value
so large that it may be ignored in
most circuit calculations.
16. Summation. After independ
ent waveshaping operations have
been performed on various signals,
combination is accomplished by sum
ming the individual results. Of the
several available methods, the simplest, shown for two inputs, uses the
resistor network of Fig. 19. From the circuit we see that the two
input currents are
e\ — e„ . ej — e
Fig. 19. Summation circuit.
li =
Ri
Rz
Provided that the output level is kept low compared with each of the
inputs, the current through each input resistor is approximately inde
pendent of the output voltage. But this is possible only if R <K Ri
and R <K R2. Almost the complete voltage drop appears across the
input resistors, and
(122)
»iS
Ri
The output becomes
ti= R t
e, = R(ii + it) =
R , R
W ei + p" e *
Hi tit
(123)
«2° ^Sl&Lr
L
14 MODELS AND SHAPING [CHAP. 1
Even when R is not small compared with the other resistors, the out
put voltage can still be found by the linear superposition of the indi
vidual terms. The circuit operates by converting the input voltage
into a proportional current and then summing all the individual inde
pendent currents. Equation (123) shows the output's proportionality
to the sum of the individual inputs, each of which is multiplied by a scale
factor. Determination of this factor is through a choice of the input
resistors Ri and R^. Summation
£ may be extended to multiple inputs
1 ' by simply adding additional input
9 branches.
. . A ~ J„ The condition for summation,
e 3 o WW ' < p 
r 3 £ i.e., the output voltage small com
pared with each input, is exactly
Fig. 110. Combined operations (differ the same as that squired for inte
entiation, integration, summation). gration and differentiation, and all
three operations may conveniently
be combined in a single circuit as illustrated in Fig. 110. The output,
expressed by Eq. (124), is found by calculating and sketching each term
individually and summing the results; this is often carried out graphically.
*»**£ +271 /** + #,•■ (1_24)
CR must be small and L/R very large compared with the time interval
of the various signals. The inputs, if desired, may be supplied from the
same source.
17. Approximate Solutions of the Singleenergy Case. The results
of the previous sections are quite valuable in that they open up two classes
of singleenergystorageelement circuits to rapid solution. The first
group consists of those with short time constants, the differentiators;
the second, those with long time constants, the integrators. These two
cases, representing only about 10 per cent of the possible range of RC or
RL circuits, include, however, a much larger portion of the important
engineering problems.
In handling a problem of this nature, our first step is to compare
the time constant with the input signal and on the basis of this comparison
to classify the circuit response. If it is a differentiator, the voltage across
the resistor or inductance may be sketched by inspection, and the drop
across the other circuit element is simply the difference between the
applied input and the now known branch voltage. In an RC integrator,
the voltage across C is the easily found quantity, with the resistor voltage
remaining the single unknown. This is illustrated in Fig. 111, where
the signal is assumed to be applied at zero time and where we shall
Sec. 17]
LINEAR WAVE SHAPING
15
apply the techniques of integration in finding the voltage across both B
and C. Because C cannot pass direct current and e« has a dc level, the
«i
1
Hf
RCl msec
10
t
e,(t)
10
20
40
60
80
(,/isec
10
f 9.8
eo(0
0?
?=—<■—
in
t
102
1
Fio. 111. Integrator circuit and approximate output.
output waveshapes sketched (Fig. 111) require a minor correction. The
voltages must be shifted slightly until both have zero average value.
Figure 112 shows the final periodic solution.
9.9
10.1
Fig. 112. Final periodic solution of the problem of Fig. 111.
It is often convenient to be able to express the response of a longtime
constant circuit to square or rectangular inputs as an equation. The
exact solution of any singleenergystorageelement circuit is, of course,
e„(t) = A + Be'*
(125)
Substitution of the series expansion [Eq. (126)] for the exponential simpli
fies the final result. Since the time constant is long, provided that we
restrict the ratio of the maximum interval to the time constant, <„/t,
16
MODELS AND SHAPING
[Chap. 1
to less than 0.1, an error of less than 5 per cent is introduced by the
neglect of the higherorder terms of the expansion.
(126)
t
Therefore, for t/r < 0.1, Eq. (125) may be approximated by
e.(0 Si+iflj)
(127)
which is recognizable as the equation of a straight line having a slope
B/t.
If the exact expression of the response [Eq. (125)] were differentiated
and evaluated at t = 0, the slope found would be the same as that
obtained from the first term in the series expansion. The interpretation
which follows is that if the circuit response is restricted in time to the
beginning of the exponential decay, this small portion of the curve may
be represented by a straightline segment.
18. Doubleenergystorageelement Systems. Many circuits con
taining two energystorage elements have poles separated widely enough
so that the transient response can be approximated by treating them as
two isolated singleenergy circuits. Of course there must be continuity
across the boundary between the two individual response curves and
they must satisfy the original system. The pole which is located far
from the origin (small time constant) will determine the circuit's behavior
with respect to any fast changes in the input excitation. It predom
inantly controls the initial rise. The pole which is located close to the
origin is related to the lowfrequency response of the system and will
contribute a slow exponential to the
output. It will determine the hold
ing power, i.e., the decay rate.
In effecting the separation of
the system response into two time
regions, we shall have to depend on
the physical characteristics of the
energystorage elements for clues as
to the permissible approximations.
This technique is best illustrated by
an example, and to this end we shall analyze the simple BCcoupled
amplifier whose incremental model is shown in Fig. 113.
Consider the application of a voltage step of height — E at the grid
of the tube. The output is constrained in its time rate of rise primarily
by C,. We see from the circuit (Fig. 113) that the full charging current
of C. together with any current through R a must also flow through C e .
Fio. 113. Incremental model for an
BCcoupled amplifier.
Sec. 18] linear wave shaping 17
If, as is normally the case, R„ is very large, the total charge flow just
after the excitation is applied is essentially determined by the uncharged
shunt capacity C,. The same charge is also accumulated in C e . Since
C c ~S> C„ the voltage across C c will r p
change but slightly while C, charges I * v ^ v
fully. Thus the coupling capacitor JL
may be assumed to be a short circuit ^ pe^jiE ^ r l R*^ C* 5 ^ 4>i
during this entire interval and the
equivalent circuit is reduced to one
containing a singleenergystorage
element (Fig. 114). Fla Uli ' Rise e l uivalent circuit 
The final steadystate output and the circuit time constant are found
by taking the The>enin equivalent across C. They are
Em1 = r p + L R L f\'R t E ri = C ' {R ° H Rl l! r *> (1 ' 28)
The parallel lines indicate that the adjacent elements are in parallel;
for example, R L  R g = R L R g /(R L + R g ). This notation will be used
throughout in an attempt to keep the circuitelement relationships
apparent in any equations which will be written.
We are now in a position to write the equation defining the initial
portion of the output response.
e„i = B»,i(l  e«"0 (129)
In four time constants, the output rises to 98 per cent of the steadystate
value E„i and the initial rise may be assumed complete.
During this whole interval, C c is charging, even though it is doing
so very slowly. The relatively large current required (because of the
large value of C c ) will now control the output voltage and swamp any
contribution from the discharge of C,. We are justified in ignoring C,
and in removing it from the circuit. If the initial value of the output
across R g , upon the sudden excitation of the system, is now calculated,
we also find it to be E Ml . As C c charges, the output decays toward zero,
with the new time constant
t 2 = C c {R g + R L  r P ) (130)
Thus the equation defining the final portion of the response may be
written by inspection.
«o2 = E, tl e l ' r ' (131)
The only question remaining unanswered is, At what point will the
decay take over from the initial rise? If we compare Eqs. (128) and
(130), we see that both the capacity and the resistance terms of n
18
MODELS AND SHAPING
Fig. 115. flCcoupled amplifier out
put — step excitation.
[Chap. 1
are much larger than n. Since the decay time constant is so very much
longer, the error introduced by starting the decay anywhere in the
vicinity of zero will be negligible. We might just as well choose the
most convenient point, the end of the initial rise, that is, 4n. The total
output is sketched in Fig. 115.
The exact solution, the dashed line, shows the greatest deviation at the
peak asymptotically approaching the approximate solution at both long
and short times. By the very nature
of the assumption made, i.e., com
plete charge before any discharge,
the exact answer will always lie
below the approximate one. As
the separation of time constants in
creases, agreement between the two
answers improves. For r 2 > IOOti,
as is normal in this type of amplifier,
agreement is extremely good.
When an amplifier is used in pulse applications, the time response,
rather than the related highfrequency 3db point, is used to characterize
its quality. Even if the amplifier transmits all signals faithfully, it may
still delay them by a fixed time interval ; this delay will also be considered
as a defining quantity. Generally, the shape of the output will be
affected, and in order to separate the rise time from the inherent delay,
we choose the 10 per cent value as our lower reference point. Moreover,
the applied pulse makes its presence known long before the output reaches
steady state; the 90 per cent point is therefore arbitrarily chosen as the
upper reference value. An amplifier's rise time is defined as the time
required for the output to rise from 10 to 90 per cent of the final steady
state voltage, after the excitation by a unit step. Substituting these
limits into and solving Eq. (129) gives the rise time of a single stage as
t r = 2.2n (132)
The smaller the time constant, the faster the initial rise rate. The
rise time ranges from several microseconds in an audio amplifier down
to a few millimicroseconds in systems designed primarily for pulse
amplification.
When the input is a square wave, the amplifier output must eventually
become periodic. The stray capacity present slows the rise of the leading
edge, and the coupling capacity introduces some tilt on the flat top.
The rise time of each half period will be the same as that given by Eq.
(132) for the unit step. However, the amount of tilt depends on the
duration of the half cycle and must be defined in terms of the wave period.
If the half period T/2 is small compared with the decay time constant
Sec. 19]
LINEAB WAVE SHAPING
19
r 2 , only a very small decay will occur over each half cycle, and this may
be approximated by a straight line. From Eq. (127), the linear repre
sentation of this region, we find that the initial value of the top is B
(A represents the steadystate value, which in this case is zero). The
final value, at the end of each half cycle, becomes B(l — Tfbn). Tilt is
defined as the relative slope of the top of the square wave and is
8 =
Ej — Ef
2t 2
(133)
S is often expressed as a percentage. Measurement of the tilt for a given
squarewave input serves as a convenient method of evaluating the hold
ing response of an amplifier, just as a measurement of the rise time char
acterizes its highfrequency response.
19. Compensated Attenuators. Attenuation of signals through pure
resistive networks generally proves unsatisfactory because of the stray
capacity loading of the output (C 2 in Fig. 116). Instead of the rapid
response to a step expected from a resistive attenuator, the output rises
with a time constant
t = (7,[(fii + R.)  Ri]
toward the final value
E„
Rz
Ri + R e ( R2
E
In order to prevent the source impedance from affecting the signal
division, usually R x S> R,; otherwise a change from one network driving
source to another would also result in a new attenuation ratio.
Fig. 116. Uncompensated attenuator. Fig. 117. Circuit of the compensated
attenuator used for the calculation of the
initial rise.
The attenuator response can be greatly improved by shunting Ri
with a small capacitor C\. Since it furnishes a lowimpedance path
to the initial rise, the output will reach steady state much sooner. At the
instant of closing the switch, the two uncharged capacitors are effectively
short circuits. Since most of the current will flow through them, rather
than through the parallel resistors, we shall remove Ri and Ri from the
circuit while investigating the initial rise (Fig. 117). The output now
20
MODELS AND SHAPING
[Chap. 1
rises with a time constant
toward
Tl
R,
E,,i =
Ci + C,
Cl E
Ci + d
(134)
(135)
By assuming that steady state is reached before the decay begins, we
can draw the model of Fig. 118 to represent the final portion of the
response. In this circuit the initial charge on each capacitor is indicated.
The output now rises or decays from its initial value toward a steady
state value E ss 2.
E M = n R * n E (136)
R\ + J?2
The new time constant becomes
r 2 = (fii  fl,)(Ci + C)
(137)
Comparison of Eqs. (134) and (137) indicates that t 2 is much longer
than n. C\ and C 2 in parallel are obviously larger than the same two
capacitors in series. R„ had been assumed small, so that the parallel
combination of Ri and # 2 will probably be the bigger term.
rW
r
Ci
e 2 (t)
=£
I
c 2
E R z $ Cz^^E e 2 (t)
Overcompensated
Fig. 118. Decay model — compensated
attenuator.
 Undercompensated
No compensation
t
Fig. 119. Compensatedattenuator out
put (slightly exaggerated for purposes of
illustration).
If the two steadystate values [Eqs. (135) and (136)] are equal,
there will only be one transient, the initial rise. By equating these two
voltages and solving, we arrive at the conditions for the optimum attenu
ator response,
R\C\ = R2C2 (138)
Compensating the attenuator greatly improves the rise time, as shown
in Fig. 119. When .R1C1 < jR 2 C 2 the attenuator is overcompensated,
LINEAR WAVE SHAPING
21
introducing an overshoot. If we are interested in attenuation of triggers
(sharp pulses), the overcompensated case is often desirable since it gives
the fastest rise and the largest initial amplitude. Much to our regret,
the .RCcoupled amplifier cannot be compensated because r p does .not in
fact exist as an entity across which a capacitor can be connected. Any
external capacity added only parallels the existing strays. This simply
increases the rise time, with a corresponding degeneration in the overall
response.
The initial rise time of the compensated attenuator is usually so short
that it may be ignored. This leads to the assumption that the output
instantaneously jumps to the point from which it starts decaying toward
the final steadystate value.
Treating the same problem from the pole and zero viewpoint and
neglecting the initial rise (R, = 0), we see that the placing of Ci across
Ri introduces a zero along the real axis. Proper adjustment moves
it into coincidence with the pole determined by R2C2, annihilating that
pole. Since the circuit no longer contains any poles, the response cannot
be of exponential form but must be constant.
PROBLEMS
11. The switch in the circuit of Fig. 120 is closed at t = and reopened at t = 10
msec.
(a) Sketch the voltage waveshape appearing across the capacitor, giving the values
of all time constants and break voltages.
(6) Sketch the waveshape of the current flowing through the switch and evaluate
the initial and final values of this current by using the information of part a. Do not
solve the exponentialresponse equation in this part.
50vS:
2K IK
•2K
Fia. 120
=t=2/*f
100 v^
Fig. 121
12. In Fig. 121, Si is closed at t  and <S 2 1 msec later. Evaluate and sketch
on the same axis the current response of both inductances. Make whatever reason
able approximations are necessary to simplify the calculations, including changing
the ideal elements to notsoideal elements which become ideal in the limit. Repeat
if Si is closed 100 j*sec after Si.
13. The capacitor of Fig. 122 is initially charged to 50 volts with the polarity
shown. The circuit is energized by closing Si at t = 0. When the output reaches
22
MODELS AND SHAPING
[Chap. 1
— 10 volts, switch Si is closed. Draw the output voltage waveshape, indicating all
time constants, and solve for the time delay between the closure of the two switches.
100 v=
Fig. 122
14. Repeat Prob. 13 if the bottom of the 1megohm resistor is returned to +200
volts with respect to ground instead of directly to ground. Si will now be closed when
the output rises to zero. All other conditions remain unchanged.
16. A rectangular voltage wave, such as the one shown in Fig. 13, is the driving
signal applied to a series RC circuit having a time constant of 10 msec. The 10volt
positive peak lasts for 30 msec, and the — 2volt negative peak for only 5 msec.
(a) Find the steadystate minimum and maximum voltages appearing across the
capacitor.
(6) What voltage would be read on a dc voltmeter connected across the capacitor?
Explain your answer.
16. The excitation of a parallel RL circuit is a square wave of current having a
peaktopeak amplitude of 50 ma and a period of 200 Msec. The inductance is 1 henry,
and the resistance 5 K.
(a) Sketch the steadystate node voltage and evaluate the maximum and minimum
values.
(6) Find the power dissipated in the resistance.
17. The periodic current flowing in the series RLC circuit of Fig. 123 is given.
What waveshape and values of excitation voltage must be applied to cause this cur
rent flow?
<1
+ 100 ma
100 ma
Fig. 123
18. A periodic triangular voltage of 200 volts peak to peak appears across the
parallel RLC circuit shown in Fig. 124.
/ 1
/ 1
f 1
l\
1 \
I
\30 40
50
/ 1 t,jUSQC
10
20
\ \
\ i
i
i y
/60 70 *
Fig. 124
LINEAR WAVE SHAPING
23
(a) Calculate the inductance that will make the peak current supplied equal to
100 ma. (The dc level of the input current is zero.)
(b) Sketch the current waveshape and give the values at all break points.
19. The nonperiodic signal shown in Fig. 125 is applied to a series RC circuit.
Calculate the approximate response under the following conditions:
(a) The voltage across C when the time constant is 10 msec.
(6) The voltage across R when the time constant is 10 msec.
(c) The voltage across R when the time constant is reduced to 10 jjsec.
t.jisec
Fig. 125
110. (o) The triangular voltage of Fig. 124 is applied to an integrator circuit
having a time constant of 25 msec. Sketch the steadystate voltage appearing
across C.
(b) If a signal consisting of only the positive portion of the triangular wave of
Fig. 124 were applied to this integrator, how long would it take for the output to
rise to 9 volts?
111. (a) Under what conditions will one of the branch currents of a parallel RC
circuit be proportional to the integral of the driving signal? Which branch?
(6) Show that a parallel RC circuit can also be used for current differentiation.
112. Repeat Prob. 11 1 with respect to a parallel RL circuit. Explain what further
restrictions, if any, must be imposed as a result of using a nonideal inductance (wind
ing resistance) .
113. (a) A 100volt symmetrical square wave having a period of 100 iiaee is applied
as the input to an integrator having a time constant of 10 msec. The output is
coupled through an ideal isolation amplifier which multiplies the voltage by a factor
of 50. The amplifier output is then passed through a differentiator having a time
constant of 1 /jsec. Sketch the ultimate output.
(6) The same circuits and input are again used except that the square wave is first
differentiated, then integrated, and finally amplified. Sketch the output appearing
in this case and explain any discrepancy between this answer and the one for part a.
114. The periodic waveshape of Fig. 16 is used as the input to the circuit of
Fig. 15.
(o) Sketch and evaluate the output waveshape under the following conditions:
t = 400 jisec, t = 200 msec.
(6) Drawing on the answers of part a, sketch, without evaluating, the output wave
shape appearing when t = 20 msec. Justify your answer without finding the exact
solution.
116. Two signals are combined in the summation circuit of Fig. 126. Draw the
output, giving voltage values under the following conditions:
24
MODELS AND SHAPING
[Chap. 1
(a) The circuit is as shown below.
(b) We replace the 10K resistor by a 0.05juf capacitor.
(c) The circuit is further modified by also replacing the 500K resistor with a
0.005juf capacitor.
100 v
t
c :
100
100 v
? 3
6 7
1
4 5
t,
msec
1 M
500 K
e 2 o WW
.10 K e„
Fig. 126
116. We are interested in generating a reasonable approximation to the signal
shown in Fig. 127. Under the conditions listed, find the circuit required and the
values of components needed. Make your smallest resistor 10,000 ohms.
(a) The only signal available is ei of Fig. 126.
(b) The only signal available is a 200volt peaktopeak square wave of the proper
period.
(c) Repeat parts o and b but use a series combination of circuit elements for the
output branch instead of a summationtype circuit.
117. A singlestage amplifier having the equivalent circuit shown in Fig. 113 is
excited by a 1volt step at the grid. The tube and circuit components are r v = 10 K,
n = 25, R L = 20 K, R, = 100 K, C, = 200 n&, and C = 0.01 fit.
(a) Find the output response, giving all time constants and steadystate values,
by the approximation method discussed in the text (time yourself).
(6) Solve for the exact response, using either classical or transformation methods,
and compare both the time required and the accuracy of solution with those found in
part o.
118. In the circuit of Fig. 128, a unit step of current is applied at t =0. Find t'i.
Sketch your answer and give all time constants and steadystate values. Justify the
approximations made in the course of your solution.
LINEAR WAVE SHAPING
25
2.2 K
T~7>\
'100
0.5 h
100 mh
Fig. 128
119. (<z) Derive Eqs. (130) and (133).
(6) Calculate the rise time of the amplifier of Prob. 117.
(c) What is the lowestfrequency square wave that would be transmitted without
introducing more than 5 per cent tilt? Sketch the output if the peaktopeak voltage
is 10 volts. Give the actual voltage values.
(d) In order for a pulse to register its presence at some later point in the system,
at least 80 per cent of it must be transmitted by the amplifier of Prob. 117. What is
the narrowest pulse that may be used, and how will it appear after passing through
the amplifier?
120. Find the output response of the circuit of Fig. 129 if the input is a 100volt
peaktopeak square wave having a period of 200 /isec and if Li is (a) 100 mh, (6)
500 mh, (c) 400 mh. Plot the three waveshapes to scale on the same graph.
5M
Xo WA,
o — '000' 1 VW '
Lj 20 K
i o
>5K
«in
k e
100 mh<
yo
1 M
100 /^f
>10K 4:0.01
Mi
Fig. 129
Fig. 130
121. The two input signals of Fig. 126 are applied to the circuit of Fig. 130.
(a) Sketch the steadystate output, giving all voltage values when ei is connected
to terminal X and ej to terminal Y.
(b) Repeat part a if a 20M/»f capacitor is inserted across the 5megohm resistor,
(e) Repeat part 6 with the two signals interchanged.
BIBLIOGRAPHY
Brenner, E., and M. Javid: "Analysis of Electric Circuits," McGrawHill Book Com
pany, Inc., New York, 1959.
Guillemin, E. A.: "Introductory Circuit Theory," John Wiley & Sons, Inc., New
York, 1953.
CHAPTER 2
DIODE WAVESHAPING TECHNIQUES
Waveshaping functions performed by purely linear circuits — differ
entiation, integration, summation, and not much else — are relatively
limited. If at some point within any operation we could, at will, change
the value of even one circuit component, there would result a tremendous
increase in the possible shapes of the output produced. For example,
if in the middle of integrating a signal, the time constant were suddenly
to be greatly decreased, then the circuit would immediately become a
differentiator, with a resultant total output that could not possibly be
produced in a purely linear circuit. When considering periodic inputs,
the component changes should be voltage or currentcontrolled, e.g., a
bivalued resistor — a high resistance at voltages below a threshold value
and low resistance thereafter, or vice versa. Otherwise, the circuit
change would not occur at the same voltage point of each cycle and a
periodic output would not result from the periodic input. One method of
obtaining this response is through the use of two resistors and a switch,
which is thrown from one to the other and back again as the circuit
voltage rises or drops past the threshold value. Of course, for fast
complex waveshapes, mechanically operated switches could not possibly
operate rapidly enough, and how could we throw the switch continuously?
Conveniently for us, there exist a number of voltagecontrolled non
linear devices with approximately the properties desired. One of the
most common of these devices is the diode, and in this chapter we propose
to examine its application to various circuits, making use of its inherent
nonlinearity.
21. Passive nonlinearcircuit Representation. The response of any
network to a forcing function is determined by the individual character
istics of the various components comprising that network. In the
analysis of socalled linear circuits, ideal elements (resistors, capacitors,
inductors, and voltage and current sources) are convenient fictions
introduced to simplify engineering calculations. The actual physical
elements are not only slightly nonlinear, but also include various para
sitics; e.g., a capacitor contains lead and winding inductance and its
capacity varies somewhat with the charge storage. Analogous char
26
Sec. 21] diode waveshaping techniques 27
acteristics may be detected in all other "linear" components. These
minor secondorder effects will cause at most a slight correction in the
calculated idealnetwork response. Their inclusion in the problem will,
however, obscure the phenomena under examination beneath the com
plexity of the additional mathematics. Thus, in the interests of sim
plicity, we are completely justified in treating only ideal linear elements.
Following a similar argument, we can postulate the existence of an
ideal nonlinearity, which we shall call the ideal diode, to account for
the response of a large class of grossly nonlinear systems. This element
has as much validity as our other circuit components and will be utilized
in networks in much the same manner. It has the bivalued character
istics expressed both graphically and in equation form in Table 21.
The ideal diode is an absolute short circuit for any positive current flow
and an open circuit when the polarity across its terminals is negative
(backbiased). Its voltampere plot consists of two connected segments
with the break between the two regions occurring at e<( = (or id = 0).
Thus the ideal diode is identical with the voltagecontrolled switch pre
viously discussed, the switching value being zero volts.
Table 21
Element
Symbol
Defining equations
Voltampere
characteristic
Ideal diode.
Resistor .
Voltage source .
Current source.
<>
ed = or id >
and ed < or id =
es = iR
i = Oe B
e, = constant = E
or e, = e(t)
i, = constant = /
or i, = i(l)
,j Slope
0/G
Table 21 illustrates the schematic representation of the branch ele
ments, with which we shall be concerned, together with their defining
equations and voltampere plots.
The combined circuit response of these elements is found directly
from Kirchhoff's voltage and current laws. Any number of elements in
series must have the same current flow, and thus the total terminal
28
MODELS AND SHAPING
[Chap. 2
voltage will be the sum of the individual drops measured at this current.
A parallel arrangement has an input current that is the sum of the
current flow in each branch evaluated at the common terminal voltage.
The diode introduces constraints dictated by its nonlinearity.
«r<«
.X
Q\>
l a
e«
e d /
/\
J
/<*
V
•'Slope 5
/
IE
e
/
/
(a)
I
7
/
/
/
/
s
I'*
(b)
Fig. 21. Two biased diode circuits and their voltampere characteristics, (a) Series
combination of elements; (b) parallel combination of elements.
Let us now consider the two examples shown below. In Fig. 2la
the series diode limits the current flow to positive values, the battery
shifts the break point to the right, and the resistor introduces a finite
slope in the conduction region. Summing the individual equations given
in Table 21 leads directly to the defining equations for the compound
branch.
c« = e d + e B + e.
e„i = E + i a R for i a > or e„ > E
(2la)
e„2 = ed + E for i a = or e„ < E
(216)
Two equations are necessary because of the bivalued diode properties:
one holds for the conduction region (2la), and the other for the back
biased zone (216). They agree at the boundary point, which may be
expressed in terms of the applied terminal voltage by setting i = in
Eq. (2la) and ea = in Eq. (216). Figure 2la illustrates the graphical
summation of the individual curves, i.e., the superposition of the element
voltage drops. We might note that the voltage drops add normally
in the positive current region but that the reversebiased diode (ed < 0)
absorbs the total loop voltage.
Sec. 21]
DIODE WAVESHAPING TECHNIQUES
29
Reversal of the diode would establish the conduction zone for negative
rather than for positive currents, and reversal of the battery would
shift the break point to the left instead of to the right. The diode
connection determines the permissible currentflow direction, and the
battery voltage only sets the breakpoint location.
In Fig. 216 the currents must be summed over the voltage range per
mitted by the diode (for negative voltages it is a short circuit forcing
ia = to + id + i,
iai = Ge a + I e„ > or i a > I (22a)
i«2 = id + I e„ = or i„ < I (226)
The shunt current source shifts the current coordinate of the break
point to a positive value as shown in the graphical characteristic of Fig.
216. Again, the diode restricts the region where graphical addition is
necessary, in this case by introducing a short circuit upon conduction.
1
1
Slope Gi i
'a
Slope fG 1 +G 2 ;
/
1
E
e o
(a) (b)
Fia. 22. (a) Combined biasing of piecewiselinear circuit; (6) voltampere charac
teristic.
Even if these circuits did not contain a nonlinear element, the inter
cepts of the seriesresistance and parallelconductance curves would still
differ from zero by the value of the voltage and current biases, respec
tively. The addition of the diode only limits the portion of the total
shifted curve which may be used.
In general, we can say that :
1. If there is no current source included across the diode, then one
coordinate of the branch break point will always be the bias voltage con
tained in the diode branch.
2. If there is no voltage source included in a parallel arrangement of
elements or branches, then one coordinate of the break point will always
be the value of the current bias across the diode.
Combining parallel current and series voltage biasing with the single
diode permits us to locate the break point anywhere in the voltampere
plane. Moreover, by including both series and parallel padding resist
ance, the slopes in the two zones may be individually controlled (Fig. 22).
30
MODELS AND SHAPING
[Chap. 2
Suppose that we are interested in evaluating the input voltampere
characteristic of any single diode circuit. How would we do so? We
could, of course, solve the problem graphically, by summing the response
of each element. For more complex circuits, such as the one shown in
Fig. 22, this process may become somewhat tedious.
&
±_ fl,E
B.+Jfe
(a)
fi 2
l< 1
X*
E
fb)
i°i
)\i
I
°'°P e r,i\rJ
<s^> — /
^ E  R i E
S l +R !
e«
(c)
Fio. 23. Models illustrating the solution of the circuit of Fig. 22. (a) Equivalent
circuit when diode conducts and the Thevenin equivalent circuit; (fc) circuit when
diode is backbiased and the Norton equivalent circuit; (c) composite voltampere
characteristic.
For an algebraic solution, the simple diode circuit may be replaced
by the two implicit resistive models, one of which holds when the diode
conducts and the other when it is backbiased. By superposition from
the model of Fig. 23o, where the current source is shortcircuited by the
conducting diode,
e " +1 ii>0 (23)
la =
Ri I Ri Ri
From the second model (Fig. 236), which holds for d < 0, we obtain
the other denning equation,
= — + J
Ri
(24)
Sec. 21] diode waveshaping techniques 31
The boundary between the regions is the intercept of the two straight
lines whose equations are given by (23) and (24). This is shown
graphically in Fig. 23c. Equating and solving for the terminal break
value eab yields
eat, = IRt E
The current break coordinate t„j can now be found by substituting e^
into either defining equation.
We might observe that the break does not occur at the intersection of
i a = I and e a = — E. This is a direct consequence of the additional
current flow through Ri (due to E) and the additional voltage drop
across Ri (due to I). The reader might remove each resistor in sequence
and observe the effect on the breakpoint location.
Each of the two models may be reduced to the Thevenin or Norton
equivalent circuit, shown in Fig. 23, by algebraic manipulation. The
opencircuit voltage or shortcircuit current source is the value of the
appropriate intercept, and the slope of the line segment is the resistance
or conductance.
Since it is known that the voltampere curve consists of two linear
contiguous segments, it follows that three pieces of information are
sufficient to define these regions. Instead of writing the complete equa
tion, we can simply solve for the coordinates of the break point and the
two adjacent slopes. While doing so, attention must remain focused on
the diode because its state controls the circuit behavior. Example 21
illustrates this technique for the circuit of Fig. 22.
Example 21. If we select the conducting state in Fig. 22 as our arbitrary starting
point, the diode will short the cun ent source and reduce the network to one contain
ing two resistors and a voltage source (Fig. 23o). This occurs where the diode cur
rent is zero and where the current through Rz equals I. Thus the break voltage is
the total branch drop at this current flow,
e„i = IR2 — E
For Ri = 10 K, Rt = 1 K, E = 25 volts, and 7 = 5 ma,
e„ h = 1 K X 5 ma  25 volts = 20 volts
The other coordinate becomes the sum of the current through R it because of the
applied break voltage e.t and the bias current /. From Eq. (24),
—20
tat, = 5 ma + „ = 3 ma
Within each region the slope is found by calculating the incremental conductance:
first when the diode is conducting (e„ > —20), and next when it is backbiased
(e„ < —20). These values are already indicated in Fig. 23c. In the forward region
it is 1.1 millimhos, and in the backbiased zone it is only 0.1 millimho.
32
MODELS AND SHAPING
[Chap. 2
We conclude that once the nonlinear circuit is represented by a model
containing a diode, we can write two sets of linear equations to describe
the system response, one equation holding while the diode conducts
and another when it is backbiased. These' equations must coincide
at the boundary point (ed = 0); the continuous diode characteristic
introduces no voltage or current discontinuity, and neither should the
model or the equation. By using two regions to represent the nonlinear
device, all theorems and techniques of the solution of linear circuits
become applicable in each region. The only complexity introduced is
that upon completion of a solution we must check to see that we have not
left one linear region and entered
another. If the circuit has done
so, then the original equation,
written for one region, will not hold
and the answer will be incorrect.
The problem must subsequently
be reformulated and resolved.
22. Physical Diode Model. A
semiconductor diode has volt
ampere characteristics of the type
shown in Fig. 24. Note the scale
change as voltage and current
become negative. Vacuumdiode
characteristics are similar, differing
primarily in their having a much larger forward voltage drop and positive
rather than negative reverse current.
Circuit analysis is greatly simplified once the actual characteristics
are approximated by the two straightline segments shown superimposed.
When drawing these lines, it is convenient to start from the origin and
intersect the actual characteristics at the center of the operating region.
However, since this is only an approxi
mation and the individual diode devi
ates rather widely from manufacturers'
data, we find any reasonable set of
lines acceptable for most purposes.
Their slope has units of conductance.
We conclude that the diode may
reasonably be represented by two con
stant resistances, a forward resistance
Tf, when ed > 0, and an inverse resist
ance r r , for ed < 0. The change from one to the other takes place
when the voltage across, and the current through, the diode drops
to zero. Thus the model formulation makes use of the ideal diode
Note change
of scale
Fig. 24. Semiconductor diode character
istic.
Qt
<fc
Fig. 25. Diode model representa
tion.
Sec. 23]
DIODE WAVESHAPING TECHNIQUES
33
as the switching element (Fig. 25) ; it shunts the very large r r with the
much smaller »y upon conduction at e<i > 0. Therefore ry and r r in
parallel are almost exactly equal to ry. If this inequality did not happen
to hold, in any particular case, we would modify the model by increasing
the series resistance so that the parallel combination is the actual forward
resistance of the diode. The capacitor shown in the model represents
the major parasitic element present.
Forwardresistance values range from a fraction of an ohm to 250 ohms,
and reverse resistance from about 25 K to 1 megohm, depending on the
particular device. The lower values generally appear in power rectifiers,
and the higher ones in generalpurpose and switching diodes. Special
purpose diodes are available having very small forward and extremely
large reverse resistances, and these very closely approximate an ideal
device.
E,r^L
(a) (b)
Fig. 26. Zenerdiode characteristic and model.
Silicon diodes designed for operation into the Zener voltage region
have the voltampere characteristics of Fig. 26a. It may be seen that
three linear regions are necessary to define properly the complete curve,
and therefore two ideal diodes are required in drawing its model (Fig.
266). Generally, the small Zenerregion resistance rz may, as a first
approximation, be taken as zero.
23. Voltage Clipping Circuits. One of the most widely used nonlinear
circuits utilizes the bivalued properties of the diode to control the signal
amplitude. In the ideal shunt clipper of Fig. 27a, when the diode is
backbiased, the shunt branch is opened and the input voltage is trans
mitted through Ri to the output, unchanged and unaffected by the
network.
On the other hand, once the rising input forces the diode into con
duction, it may be replaced by a short circuit. The subsequent output
is E, provided, however, that it is exceeded by the input. Thus the
two zones of operation are, first, a transmission region where the output
equals the input (e„i = e ln ) and, next, a clipped region where the output
34
MODELS AND SHAPING
[Chap. 2
is determined solely by the bias (e.,2 = E). These zones are plotted as
the ideal transfer characteristic in Fig. 28.
Turning now to the physical diode clipper shown in Fig. 276, the
addition of r r and jy establish information transmission paths which are
nonexistent in the ideal circuit. In each of the operating regions only
one of the diode resistors need be considered and the actual output may be
oWV — 
L
1
t
I 1
+ e <
_
L
o WW
X
r r >e d
1
(a) (b)
Fig. 27. Voltage clipping circuit (shunt diode), (a) Ideal model; (6) physical model.
Fig. 28. Clipper transfer characteristics.
found by the linear superposition of the contributions from both voltage
sources. When c u < E and the diode is backbiased,
e i =
r,
«i
r r
e iB +
Ri
Ri + r r
E
(25)
When conducting (e ln > E), the diode may be replaced by its forward
resistance and the output voltage may be written
Co2
Ri
Ri + r,
E +
r/
Ri + r,
(26)
Both equations, each defining the operation in an individual region,
are straight lines, as shown in Fig. 28. At e in = E, the break voltage,
both Eqs. (25) and (26) give the output as E, proving the continuity
of the transfer function.
Sec. 23] diode waveshaping techniques 35
A plot of Eqs. (25) and (26), the transfer characteristic of the clipper,
appears in Fig. 28, with the ideal characteristic also presented for com
parison. Examination of Eq. (25) indicates that within the trans
mission region, the finite diode reverse resistance results in an undesirable
contribution to the output from the bias E. This contribution is the
second term of the equation. In the clipping region, the second term of
Eq. (26) represents the output contribution from the now undesired
input. We are faced with the necessity of optimizing this circuit, helped
in this by our knowledge of the behavior of the ideal circuit.
Freedom of choice is limited to selecting an optimum value for Ri.
Rewriting Eqs. (25) and (26) in the form of (27) and (28) will aid in
formulating the problem.
Col =
ET7,«( 1 + *3 (2 " 7)
■^ J! ( 1+ J:i) (2  8)
Ri
Ri
Ri + r,
The term outside the parentheses in both Eqs. (27) and (28) consists
of a constant multiplying the signal producing the desired output infor
mation in the several regions. We might consider the term inside the
parentheses as representing the desired output (unity) plus a proportional
error term. The error in both equations includes the ratio of the signal
producing the undesired transmission to the signal producing the desired
transmission. Equation (27) holds when the diode is nonconducting,
and here this ratio is E/e ia > 1 . In Eq. (28) , defining the system response
for diode conduction, the ratio of e in /E is also greater than or equal to
unity. Since the complete second terms of both equations represent
the proportional error, they should, for optimum response, be made small
compared with unity. The conditions on Ri are contradictory: in
Eq. (27) the optimum value becomes Ri = 0, and in (28), Ri = °°.
We shall define the two error terms in the parentheses as
«ol
and £02
r r VWi
Ri\Ej 2
To resolve the contradiction, "a" compromise choice for Ri will be
designated the optimum value. One possibility is a resistance that
results in error terms of the same magnitude in both equations when the
ratio of the signal producing the undesired output to the signal producing
the desired output is the same, or «<,x = «„ 2 when
U)i (£)«
36 MODELS AND SHAPING [CHAP. 2
Consequently, the compromise value becomes
Ri = V^r (29)
Substituting Eq. (29) back into the proportionalerror terms yields
We conclude that for clipping circuits a diode with the highest ratio
of inverse to forward resistance represents the best choice. In any
individual circuit, the series resistance used may not be the one found
above; particular constraints may dictate a value closer to zero or to
infinity.
+ Ri +
o — AV
X
e in
^
(b)
Fig. 29. Negative clipper transfer characteristic and circuit.
The use of the Zener region of the diode in addition to the forward
conduction region sets two clipping values instead of one. One point
is located at e d = 0, and the other at e d = E z , as indicated in the volt
ampere characteristics (Fig. 26). Biasing this diode shifts both break
points in the same direction while still maintaining their original separa
tion. It follows that one difficulty facing us is the lack of individual
control over the slopes in the two conduction regions. Any series resist
ance added contributes equally to the Zener region and the diode forward
current region. Generally, for versatility of adjustment, two individual
diodes are preferred to one Zener diode. However, the constant voltage
drop in the Zener region does avoid the necessity for inserting a separate
bias source.
When the diode in Fig. 27 is reversed, the clipping region is below
the break point; the diode now conducts through its forward resistance
Tf until e in > E. After conduction ceases, a transmission path through
the reverse resistance still exists, and the final result is the transfer
characteristic given in Fig. 29. Here the signal is clipped below the
bias voltage and transmitted above. Reversing the polarity of the bias
Sec. 24] diode waveshaping techniques 37
voltage E only shifts the break point of the transfer characteristic to
the third quadrant. We conclude that the direction of the diode connec
tion determines whether clipping will be below or above the bias value and
that the bias voltage determines the actual clipping point.
Series clipping, a possible circuit variation, gives results similar to
those found for the previously discussed shunt clipping. Insertion of the
diode in series with the transmission path (Fig. 210) allows transmission
while it conducts, in this case for
Cm < E. When backbiased, the +_
diode introduces a high series im f
pedance as the means of limiting I ,
the signal output. In this respect i + 1
it behaves just opposite to the shunt  T
iH^Wv
clipper, which stops transmission r ^ X 
upon conduction. The transfer char ^ 210. Series clipper circuit.
acteristic of the circuit of Fig. 210
is of nearly identical shape with that of the shunt clipper of Fig. 27
(differing in the slopes), and the optimum value of Ri will also be the
same as that discussed above.
24. Single Diode and Associated Energystorage Element — Some
General Remarks. In the solution of circuits containing both ideal
diodes and energystorage elements, notice must always be taken of the
immediate past history as well as of the present excitation. The storage
element should be considered as a memory which contains information
stored in the past and which permits history to play a role in determining
the present and future state of the circuit.
Our mode of attack follows the principle of timezone superposition
developed in Chap. 1. First the initial state of the circuit is determined;
next the network is solved for the time at which the diode will change
state.
Thereafter we simply treat the new circuit with the new time constant
and initial conditions. The energystorage element adds constraints to
the continuity of current or voltage across the circuit and/or excitation
changes and in this manner contributes to the new initial values.
Example 22. We shall now consider the problem posed in Fig. 211, where the"
initial voltage across the capacitor is 100 volts (its polarity is indicated in the sketch)
and where the negative 50volt 1msec pulse is applied at t = 0. The capacitor
charge remains constant across the instant of pulse application, and thus the output
drops from —100 to —ISO volts. Since the diode remains backbiased, the voltage
immediately starts rising toward +300 volts with a time constant of ti = 1 msec
(0.01 pi X 100 K). The defining equation becomes
e.i = 300  450e" r i
38
MODELS AND SHAPING
[Chap. 2
■50
0.01 fd
Fig. 211. Diode and capacitor circuit for Example 22, showing an input and the
resultant output waveshape.
Eventually the output reaches zero, the diode conducts, and the circuit changes.
Solving the above equation for the time at which e„i = 0,
(, = ti In 45 %oo  0.405 msec
In the new circuit the output voltage will charge toward the TheVenin equivalent
value of
j^ 300 6 volts
Eth —
with a new time constant t% = 20 ^sec (2,000 ohms X 0.01 ^f). We can assume
that the circuit reaches this value in 80 /usee and that it will remain there until the
excitation again changes.
At the termination of the pulse (I — 1 msec), the input rises by 50 volts and this
same jump is coupled by C to the output. Since this does not change the diode state,
the output recovers to 6 volts with the short conduction time constant of 20 /xsec.
The complete waveshape is shown in Fig. 211.
If a capacitor were placed across the diode, then the charge stored in it
would maintain the diode state invariant for some time after the excita
tion change. An inductance in series would produce a similar effect by
sustaining the current flow. We conclude that the location as well as
the type of energystorage element employed directly influences the non
linear behavior of a singlediode system. In any analysis great care
should be taken to see that the con
trolling factor, the diode state, is
always kept in view.
Some practical widely used circuits
employing various energystorage ele
ments will be discussed in detail in
the sections below (25, 27, and 29),
and they will further illustrate the
modes of solution.
25. Diode Voltage Clamps. The
function of the diode clamp of Fig. 212
is to shift the input voltage in such a manner that the resultant
maximum value of the output will be maintained at zero, without at
Fig. 212. Diode voltage clamp.
DIODE WAVESHAPING TECHNIQUES
39
Sec. 25]
the same time distorting the waveshape. The output voltage is then
said to be clamped negatively at zero. Sketches of typical input and
output waves appear in Fig. 213.
E,
cm
<i
E,
E
h h
■
«2i*
+T 2 +
E
1
t
i
i
i
i
i
i
^
Tr,C
1 < '
> 1
! J I
i
i i
i
l
i
t
t
Fig. 213. Idealdiode clamping waveshapes.
In the initial discussion we shall idealize the clamp at least to the
extent of assuming that r r = °o . Upon application of the input signal,
C charges to the peak positive value of the input with the time constant
CV/. Provided that this time constant is sufficiently short, charging will
be completed within the first cycle; otherwise it may take several cycles.
Upon completion of the charging, the voltage across the diode becomes
e d =
= e ia  E e = e in  #i
(210)
The conclusion drawn from Eq. (210) is that the output will always
be less than or equal to zero since the voltage across the capacitor, E\, is
the maximum positive value of e in . Diode conduction ceases once C
is fully charged with the diode subsequently remaining backbiased.
The output never exceeds zero, and it is said to be clamped there. For a
rectangularwave input, the clamping action is illustrated in Fig. 213
The diode serves first to introduce a lowresistance path for charging,
and then a veryhighresistance one, preventing the discharge of C.
Since the output is simply the sum of the capacitor and the input voltage,
the signallevel shift in the clamp is provided through the charge accumu
lated in the capacitor during the initial buildup. The behavior is the
same as if a battery whose voltage is dependent on the peak positive
value of the input replaces the capacitor in the series transmission path.
40
MODELS AND SHAPING
[Chap. 2
During the interval that the diode is backbiased, C discharges slightly
through the finite value of r r with the long time constant Cr T . The
energy dissipated in r r is supplied from the charge previously stored, and
of course it must be replaced once the diode again conducts. The dis
charge introduces tilt over the negative portion of the cycle. Since
the full change of input voltage appears across the diode (because of
continuity of charge in C), the output becomes slightly positive when the
input voltage rises by E at the end of each cycle. Therefore the output
appears as shown in Fig. 214. Note that the signal has become some
what distorted because of the cyclical charging and discharging.
Ti=r,C
 T 2 =r r C
Fig. 214. Physicaldiode clamping waveshape.
If the recharge time constant is excessively long, the circuit will not
recover to zero within the recharge period. A solution for this case
would follow the discussion of Sec. 13, with, however, different time
constants used over each portion of the response.
Some general conclusions as to the circuit behavior may be drawn by
realizing that for a periodic solution, the net change in the charge stored
in C over any cycle must be zero. The charge flowing into the capacitor is
Q+  f Tl i dt = [ Ti ?±
Jo Jo r,
dt
(211)
where e+ is the timevarying voltage across the output when the diode
conducts. The charge flowing out of the capacitor is
Jh r r Jo r r
dt
(212)
where e_ is the voltage across the output when the diode does not conduct.
Since there can be no net change in charge over the cycle,
Q = Q+
Equating (211) and (212) and solving for r f /r r yields
f T ' dt
r t — J° 6+ — net positive area
r r f T * e _ fa net negative area
Jo
(213)
Sec. 25]
DIODE WAVESHAPING TECHNIQUES
41
Equation (213) leads to the conclusion that for proper clamping as well
as for clipping, the diode with the maximum ratio of reverse to forward
resistance performs best (least distortion). Observe that in deriving
this equation the only assumption made is that the input signal is periodic.
Even though the waveshape used to illustrate the clamp's behavior is
rectangular, its shape appears neither explicitly nor implicitly in the
charge equations (211) and (212). The final result, that the ratio of
areas equals the ratio of diode resistances, must be independent of the
applied waveshape.
C
— It—
X
iov:=:£
Fig. 215. A triangular wave clamped positively at —10 volts, illustrating distortion.
Reversing the diode clamps the input signal positively, instead of
negatively, at zero volts, with the capacitor recharge showing itself as a
slight negative excursion. Biasing this clamp similar to the clipping
circuit sets the output voltage at any desired level, diode direction
determining positive or negative clamping. Figure 215 shows a tri
angular input clamped positively at — 10 volts, with the distortion exag
gerated for purposes of illustration.
One necessary condition for properly biased operation is the presence
of the reverse diode resistance, which establishes a path for the initial
charging of C to the bias voltage. Consider the behavior when using an
ideal diode (r r = °o) in the circuit of Fig. 215. If the negative peak
of the input voltage never reaches — 10 volts, the diode always remains
backbiased and, since C is uncharged, the input appears at the output,
unchanged and undamped. Signals with negative peaks greater than
42
MODELS AND SHAPING
[Chap. 2
— 10 volts forwardbias the diode on the peaks, and it provides the
charging path. However, once the diode has a finite value of inverse
resistance, C charges through this, and as a consequence the clamping
action becomes independent of the signal amplitude.
Referring back to Eq. (213), we see that for optimum clamping with
any given diode, the input signal should be clamped so as to minimize
the net area on either side of the clamping bias voltage. When the area
where the diode is backbiased is large, then the distortion area during
conduction will also be large.
15
i°
15
0.5
5 msec
(a)
Fig. 216. Clamping waveshapes, (a) Input waveshape; (b) output clamped nega
tively at zero; (c) output clamped positively at —30 volts.
The waveshape of Fig. 216a must be clamped so that its maximum
positive excursion is zero. It may either be clamped negatively at zero
(output in Fig. 2166) or positively at 30 volts (output in Fig. 216c).
The diagonally shaded areas in Fig. 2166 and c indicate the regions
where the diode is backbiased, and the solidly shaded areas, the regions
of diode conduction. Since, from Eq. (213), the ratio of areas in the two
cases is the same, clamping the longest portion of the period, rather
than the shortest, is preferable. This corresponds to the case shown in
Fig. 216c, which has minimum areas and hence least distortion. But
if the signal amplitude varies, then no choice exists within the constraints
imposed (maximum positive excursion limited to zero) and the signal
must be clamped negatively at zero (Fig. 2166).
Sec. 25] diode waveshaping techniques 43
Example 23. Suppose that in the clamp used with the waveshapes of Fig. 21 6a,
r/ = 100 ohms, r, = 1 megohm, and C = 0.1 id The two time constants of interest
axe
ti = r r C = 0.1 sec and n  r t C = 10 /jsec
Since ri is so very much longer than either portion of the cycle, within the backbiased
region the circuit may be treated a3 an integrator. The output thus decays linearly
toward the clamping level E.
(a) When the signal is clamped negatively at zero (Fig. 2166), the change in the
output voltage over the 5msec interval becomes
Ae 0l =^5X 10» = 1.5 volts
At the end of this period the output has risen from 30 to —28.5 volts. The input
now jumps by 30 volts, driving the output to a positive peak of 1.5 volts, from which
it recovers with the fast time constant t 2 .
(b) If the output is clamped positively at —30 volts (Fig. 216c), then, within the
interval where the diode is backbiased (0.5 msec), the output decays by only
30
Ae„ 2 = ^j 0.5 X 10' = 0.15 volt
Thus during the recharging time the maximum excursion below the — 30volt level
is 0.15 volt. The difference between these two cases is directly proportional to the
duration of the nonconducting portions of the cycle, and we have verified our original
argument.
Looking into a clamp, the signal source sees a different complex
impedance during diode conduction than when reverse diode current
flows. If the source has any internal impedance, the loading by the
forward resistance of the diode introduces additional waveshape dis
tortion. The total recharge voltage excursion divides proportionately
across the source impedance and the diode forward resistance. Since
the output is taken only across r f , the voltage in this region will be smaller
than expected, as seen in Fig. 215. During recharge, the forward voltage
drop at the output, measured from the bias value, becomes
E+ is the peak voltage drop across the total series resistance while the
diode conducts.
In addition, when using Eq. (213), the source resistance must be
added to the diode forward resistance and may even be the predominant
term. We can see that this large increase in resistance decreases the ratio
of areas, with correspondingly poorer clamping. When the diode is
backbiased, the output is also attenuated slightly because of R,. But if
R, were large enough to have an appreciable effect, it would so distort the
conduction signal that the clamper would be unusable.
44
MODELS AND SHAPING
[Chap. 2
For clamped triangular and sinusoidal signals (Figs. 215 and 217),
we are often interested in knowing the diode conduction period. Because
the source impedance of the input generator only causes division of the
recharge voltage drop, a logical first step in the calculation is to lump
all forward resistors and ignore the voltage division. Furthermore, in
a closetoideal circuit the waveshape distortion also may be neglected:
the clamp is treated as a device which only shifts the output waveshape
to a point where it satisfies Eq. (213). The two areas can now be calcu
lated in terms of the unknown conduction angle, and their ratio equated
Fig. 217. A negatively clamped sine wave.
to the known ratio of forward to reverse resistances. For the triangular
wave of Fig. 215, neglecting distortion, the area above the clamping
voltage is
Ax = HTx(Ex + 10)
(214)
(215)
The area below, measured from the bias voltage, becomes
Ai = y^T^Ei + 10
But from the input wave
Ex  Et = 40 Tx + T 2 = T (216)
In addition, from the geometry, by setting up the ratio of similar triangles,
Tx/2 T/2 _ T 2 /2
T/2
40
Ex + 10
T/2
40
\E, + 10
(217)
Substituting first Eqs. (217) and then (216) into Eqs. (214) and (215)
and equating to the ratio of resistance,
r t + R.
r r
2V
TV
(t  T*y
(218)
T 2 is the only unknown, and this equation is readily solved. We reject
the negative root because it has no physical meaning. Next E2 may be
found by substituting back into Eq. (217). Using the resistive voltage
Sec. 26]
DIODE WAVESHAPING TECHNIQUES
45
Positive
clamping
slope=l
Negative
clamping
Fig. 218. Clamper transfer characteris
tics.
divider of R. and r f and recognizing that E t is developed across both
resistors finally gives E' t .
Maintaining the clamping level in the face of changes of inputsignal
level requires fast charging of C for increasing signals and fast discharge
on decreasing inputs. Both conditions are satisfied only with a small
capacitor. However, too small a value allows excessive discharge during
the clamp cycle with resultant waveshape distortion.
Occasionally, clamps are used in systems (such as television) where
they must respond to the periodic signal but not to any noise pulse which
might also be present. Since we do not wish the output to shift appreci
ably during the narrow noisepulse
interval, we must choose a large
capacity and accept the conse
quences of a slow circuit response
and incomplete recovery within the
diode conduction region. Or, as an
alternative, the reverse resistance
might be reduced so that the effects
of the noise pulse will be rapidly
dissipated in the relatively in
efficient clamp. In a practical
circuit a compromise value of r r
and C would be chosen, satisfying as far as possible all requirements,
weighing them in order of importance.
The transfer characteristic of a clamper is given in Fig. 218. Note
that it is a straight fine of unity slope, with the locus of the end point
the bias voltage line. The y intercept depends on the inputsignal
amplitude. This is obvious if we consider that for positive clamping
the output must always (neglecting the capacitor recharge) be equal to or
greater than 2J M „. AEi is the peak negative voltage applied to a positive
clamper, and AEz is the peak positive voltage applied to a negative
clamper. These transfer characteristics have no value or meaning
except for periodic inputs. When presented on an oscilloscope, they do,
however, show whether the clamp is operating properly.
26. Current Clippers. We find the concept of duality extremely help
ful in developing the circuits used for current clamping and clipping.
By simply taking the dual of a voltage clipper, we arrive at the circuit
of a current clipper (Fig. 219a). Just as loop equations were ideally
suited for the analysis of voltage circuits, the node equations are ideal for
treating the currentactivated devices. To complete the dual relation
ships, where previously the break point of the diode piece wiselinear
model was taken at e d = 0, now it might well be regarded as occurring
when id = 0. Figure 2196 shows the transfer characteristics of the
46 MODELS AND SHAPING [CHAP. 2
current clipper. Except for the change in axis (current instead of volt
age), it is identical with that given in Fig. 29 for the voltage clipper.
Introduction of a bias current I b sets the break point of the transfer
characteristic wherever desired.
(a) (b)
Fia. 219. Current clipper and transfer characteristics.
Analysis of this circuit starts with the writing of the node equation
id = ii — It ~ is (219)
When id > 0, the diode conducts. From Eq. (219) we see that the con
duction region corresponds to ii > I b and the nonconduction region to
i\ < lb. The break point turns out to be, as expected, the value of the
bias current. Since the regions of conduction and nonconduction are
known, the two equations defining operation are, by superposition,
lol —
io2 =
h +
R
R + r r *° • R + r r
R . r,
R + r/ 1 "+" R + r,
(220)
(221)
Equation (220) holds when the diode is nonconducting, and (221) when
it conducts. The first term on the righthand side represents the desired
transmission, and the second term, the error component of load current,
i.e., the transmission of undesired current to the output. These equa
tions are analogous to the set used in describing the operation of a voltage
clipper [Eqs. (25) and (26)]. Therefore identical reasoning leads to the
same optimum value of the shunt resistor R as that previously found for
Ri in Sec. 23.
R = \/»V7
Reversing the diode changes the clipping characteristics, now allowing
conduction for i\ < I b and clipping when ii > I b .
27. Current Clamps. In the treatment of voltage clamps (Sec. 25)
we saw that the prime constituents of a clamp are an energystorage
element supplying the additional energy (previously stored) necessary
in shifting the signal level, and a diode providing short storage (charge)
Sec. 27]
DIODE WAVESHAPING TECHNIQUES
47
and long decay time constants. When we must clamp current wave
shapes, we logically turn to the inductance as our storage element. This
circuit, the dual of the voltage clamp of Fig. 212, appears in Fig. 220.
Since its basic behavior is independent of waveshape, the excitation
chosen might as well be the simplest, e.g.,
a current square wave. For the initial
discussion, assume its application at
t = as shown in Fig. 221.
The load current may, by writing the
node equation, be expressed as
LWwvvJ
to
id
ti
tL
(222)
Fig. 220. Current clamp.
During the first half cycle of the input
square wave, the diode conducts and, assuming r f = 0, the full input
current flows in the output lead. However, in the second half cycle,
ii < and forward conduction ceases. The input current must flow
somewhere. Since there was zero initial current in L, the only place left
for the current to flow is through the diode's inverse resistance. Inductive
current starts building up toward
— /„ with a corresponding decay in
the reverse diode current. Buildup
is rapid because of the small time
constant,
L
Tl = —
r r
For a sufficiently long half period
compared with n, that is, T/2 > 4t\,
charging will be completed within the
half cycle, otherwise within several
cycles. The input again becomes
positive and, with no circuit dissipa
tion, the inductive current remains
at I m . From Eq. (222), the out
put subsequently becomes
to = i\ — ( — Im)
But since t'i takes on only one of two
values, ± 7„, i„ may be either zero
or positive but never negative. The output current is composed of two
components. On the positive peak the contribution of I m , from the
signal source, and the additional contribution of I m , previously stored
in the inductance, add. On the negative peaks these components sub
Im
i — .
H
Im
t
1
l>
1
1
1
t
Im
l
rrrrs — y*^
2/m
I l
1
l
TZ
\Im
Im
Y
K
Y
t
Fig. 221. Currentclamp waveshapes.
48 MODELS AND SHAPING [CHAP. 2
tract. We have thus clamped the output current positively at zero; it
varies between zero and 27„.
The inductance acts as the circuit's memory, remembering always
the peak value of the input excitation. Once this information has been
stored, the output will be shifted to satisfy the circuit conditions estab
lished by the diode.
Physically, 77 > 0, resulting in an inductivecurrent decay during diode
conduction; this decay corresponds to the energy dissipated in the for
ward resistance. Since the decay time constant X2 is so very long, the
load current droops only slightly.
L
T2 = zr
rf
We must periodically restore the dissipated energy in the inductor's
magnetic field. On alternate half cycles (negative input), recharge
takes place, manifesting itself as a small negative excursion of output
current. The final periodic response is sketched in Fig. 221. These
results are exactly analogous with those obtained in Sec. 25 for the volt
age clamps. However, whereas recharge in the voltage clamp took place
during diode conduction, here it occurs when the diode is backbiased.
The same condition of a large ratio of diode reverses to forward resistance
is necessary for proper current clamping, and the signal will shift until
the ratio of the areas equals the ratio of diode resistance.
Bias currents may be introduced, in parallel with the diode, shifting
the clamp point wherever desired. As with the voltage clamp, reversing
the diode reverses the direction of clamping. The nonideal inductance
has associated coil resistance, which adds to the other circuit resistance
and decreases both n and t 2 , having the most pronounced effect in short
ening the long discharge time constant.
28. Arbitrary Transfer and VoltAmpere Characteristics. Many
devices and systems exhibit grossly nonlinear responses which, for
analysis, we should like to approximate by an equivalent network of
biased diodes and resistors. These nonlinearities may inadvertently
arise from the physical properties of the device and will thus represent
an undesirable characteristic. Alternatively, they may be deliberately
created and inserted in the signal transmission path in order to perform
various operations. In both cases generally more than two linear seg
ments are necessary for a satisfactory piecewiselinear representation.
This simply means that additional diodes must be incorporated into the
network — each break point corresponds to change of state of an ideal
diode — from conduction to cutoff, or vice versa. One simple example of
arbitrary transfer and voltampere curves and the equivalent diode
model appear in Fig. 222.
Sec. 28]
DIODE WAVESHAPING TECHNIQUES
49
If the network contains at most two or three diodes in a relatively
simple configuration, then, in solving for the break coordinates of any
one diode, we can always assume the condition of the others. In Fig.
222, we may assume D\ and D 2 conducting and solve for the break
values of _D 3 . Any inconsistency arising when checking the answer (e.g.,
a negative current flow through a diode that was assumed conducting)
means that the problem must be resolved from the new starting point.
(a)
i , tin m a
8 15 37.5
i volts »
(e)
Fig. 222. An example of a threediode network together with the transfer and volt
ampere characteristics, (a) Threediode circuit, (b) transfer characteristic; (c)
input voltampere characteristic.
Three diodes afford eight possibilities as to the circuit condition, but these
would be halved by the particular diode and biasing arrangement. With
more than three diodes, in other than a trivial circuit, the trialanderror
method becomes unreasonably tedious and we must seek a more organized
method of solution.
Example 24. In the case where we can readily identify the diode states, as in the
circuit of Fig. 222o, we reduce the complex circuit to a number of simpler models.
Each region is denned by the resistive network obtained when all backbiased diodes
are replaced by open circuits and the conducting ones by short circuits. If these
regions are selected by assuming the diode states, then each model must be checked
for consistency. In the circuit considered, the region limits are easily specified in
terms of the output voltage: Di conducts when e„ < 10, D% conducts when e„ > 25,
and D t is forwardbiased when e t < eu
50 MODELS AND SHAPING [CHAP. 2
Region I:
Di and Z>» backbiased (open circuits)
Di conducts (short circuit)
  *'*, ** 10 8 volts
" 01 R 1 + R t ' 1K + 4K
This output remains constant at this value until D s conducts once e^ > 8 volts.
Region II:
Di and D% conducting (8 < e„ s < 10)
Z)j backbiased
*•  g,\+ ft  + *&+*, **  ° 286e, ° + 5  7
Region III: Di conducts (10 < e». < 25)
Di and Ds nonconducting
e »a "" d — I — 5" e '"> = 0667ein
«i r "4
Region IV: Ds and D t conducting (25 < e„«)
Z>i backbiased
 _ *! II «4 . . + ?'_)'*« ft  0.4 eill + 10
Cot R,  B, + R, Cm T fl»  R,+Rt
The limits of the output are inserted into the individual equations to find the limits
expressed in terms of input voltage. These are indicated in Fig. 2226. Note that
in each region the transfer characteristic is a straight line with a slope found from the
incremental model (all battery voltages set to zero).
A further application of the type of circuit shown in Fig. 222a comes
from recognition that its input voltampere characteristic (Fig. 222c)
changes as diodes start and stop conducting. The inverse slope of these
characteristics is simply the resistance seen looking into the input
terminals. Therefore diode circuits are useful in developing almost any
required nonlinear resistance characteristics. Piecewise approximations
result in a discontinuous rather than continuous incremental input
resistance, but by taking enough segments we can approximate any
continuous resistance as closely as necessary.
A General Approach. Although we are developing more explicit
methods of analysis and the implicit techniques of synthesis of multiple
diode circuits, we shall be forced to restrict the discussion to a single
reasonably general network. This is essential if the treatment is to be
kept within bounds; the model used in describing an arbitrary nonlinear
characteristic is not necessarily unique, and therefore many feasible
solutions exist. It is neither possible nor practicable to examine all of
them.
Sec. 28]
DIODE WAVESHAPING TECHNIQUES
51
Each branch of the general network of Fig. 223a consists of a parallel
combination of biased diodes and resistors as shown in Fig. 223b. The
overall response may be determined by noting that:
1. The number of break points in the transfer or input voltampere
characteristic equals the number of diodes contained in the two branches.
2. The piecewiselinear response is completely defined by finding the
coordinates of the break points and the slopes of the two unbounded
segments.
3. The maximum possible transfer slope is unity since the network is
completely dissipative. (We restrict R to positive values.)
«12
5
«2
(a) ~. ■ (b)
Fig. 223. (a) General twobranch network; (6) typical branch configuration.
In calculating these break points, each branch may be considered
separately and then their individual responses combined. Under the
rules developed in Sec. 21, the voltage break values of any branch are
simply the bias voltages appearing in series with the internal diodes.
Once these voltages are known, the corresponding current values are
readily found by direct evaluation. It is important to determine which
diodes are operative in any region, since each one that starts to conduct
introduces additional shunt resistance and each one that stops conducting
reduces the number of elements in the branch.
As an aid to our inconsistent memory, the behavior of each branch
and the state of each diode should be tabulated as a function of the con
trolling terminal voltage. This not only enables us to construct the
response characteristics in the most organized manner, but in itself
reflects the many resistive models inherent in the diode network.
The input voltampere characteristic is the totality of the branch
responses. Under the series arrangement shown, the current must be
continuous and the voltage drops simply add. The breakpoint current
coordinates comprise the set of those found separately for the two
branches. The corresponding voltage coordinates must be found by
the direct evaluation and summation of the drops at these known cur
rents. For one branch the drop is identically the break value, and for the
52
MODELS AND SHAPING
[Chap. 2
other it may be determined from the graph or from the equivalent
resistive network.
Finally the overall transfer characteristic is obtained by crossplotting
the output branch voltages corresponding to the input break values. In
each region the slope is identical with that found from the purely resistive
network holding. Voltage sources only shift the location of the seg
ment; they cannot alter its slope. To reiterate, the complete network
response is determined by the location of the break points and the incre
mental slopes of the line segments terminating on them. Only this
information is necessary for a complete description of the network.
Example 26. The circuit of Fig. 224 utilizes series and parallelresistance pad
ding of Zener diode for its series branch and two individually adjustable diodes for its
Fig. 224. Circuit to illustrate the method of analysis of multiplediode networks.
Co> (b)
Fio. 225. Individual branch characteristics for the circuit of Fig. 224. (a) Series
branch; (b) shunt.branch.
Sec. 28]
DIODE WAVESHAPING TECHNIQUES
53
shunt branch. The voltampere response of each branch may be tabulated with
respect to decreasing terminal voltage as indicated in Tables 22a and 6. Their
characteristics are also presented graphically in Fig. 225o and 6.
Table 22a. Series Branch Response
Diode states
ei2
i, ma
Incremental resistance*
T12, kilohms
en >
10 < e 12 <
10
10 > e»
0.5
4
Di off, D 2 off
20
4
Table 225. Shunt Branch Response
Diode states
e 2
i, ma
Incremental resistance *
r s , kilohms
e 2 > 10
10
10 < d < 10
10
e s < 10
3
1
2 5
D a on, Dp off
5
D a turns off f
C.oJ.Djjoff
10
* The incremental resistance of each region is simply the parallel combination of all
resistors inserted by the diodes conducting in that region.
f The individual branch break points, which are found first, serve as the critical
points of the table. The voltage values are determined by inspection of the branch,
and the current by direct evaluation.
Now that the response of each branch is known, we are in a position to find the
total input voltampere characteristic. This can be done by direct summation of the
two graphs of Fig. 225a and 6 or by considering the data presented in Table 22.
We can construct a new table by interleaving the known break currents in descending
order (Table 23). These represent the totality of break points and separate the
linear input regions. Furthermore, the incremental resistances previously found are
in series and may be directly added. The reader should compare the appropriate
columns of Tables 22 and 23.
The only data still unknown are the values of the input voltage coordinates. For
these we must find the voltage drops across the individual branches at the known
current values. The derived voltages are given in the parentheses in columns 3 and
4 of Table 23. They may be found by linear interpolation from the tables of the
individual branch response, from the branch graphics, or directly from the circuit.
The voltage break value may also be determined by writing and solving the equa
tion of a straight line holding within each region. In the region defined by the bot
tom line of Table 22a, the segment is a straight line of slope ^K passing through the
point (0.5 ma, 10 volts).
54
models and shaping
Table 23. Total Circuit Response
[Chap. 2
Region
(1)
Conducting
diodes and
diode chang
ing state
(2)
i,
ma
(3)
en*
(4)
e 2 *
(5)
ei = ei2 + e 2
(6)
Incre
mental
resist
ance,
kilohms
(7)
Trans
fer
slope
I
II
III
IV
V
Di, D a , Dp on
Dp turns offf
Di, Da on
Di turns offf
D a on
Di turns onf
D a , Dt on
D a turns offf
D 2 on
3
0.5
1
(12)
10
(12)
10
(5)
(7.5)
10
«i > 22
22
5 > ei > 22
5
17.5 > ei > 5
17.5
22 > ei > 17.5
22
d < 22
6.5
9
25
9
14
2.5/6.5
5/9
5/25
5/9
10/14
* The derived values are enclosed in parentheses,
t Circuit break points.
Fig. 226. (a) Input voltampere response; (6) transfer characteristics of the circuit of
Fig. 224.
Sec. 29] diode waveshaping techniques 55
Thus
en(i) = 10 volts + (i + 0.5)4 K
At the known break current of — 1 ma, this yields
eit(l) = 12 volts
Similar equations may be written for the other unknown terms. Figure 226a is the
plot of the complete input voltampere characteristics.
The transfer curve becomes immediately apparent if we recognize that the incre
mental transfer slope of each region is
Aei ri 2 ( r 2
where rj and ru are the incremental resistance in the appropriate regions. Table 23
also includes these various slopes. Figure 2266 shows the crossplotting of the input
and output voltage coordinates of the break points leading to the network transfer
characteristic. The regions are those given in Table 23.
We can draw the following general conclusions from the above example:
1. Each time a series diode starts conducting, it reduces the net series
resistance and increases the slope of the transfer characteristics. Each
time a series diode stops conducting, it reduces this slope.
2. The conduction of a shunt diode reduces the branch resistance and
decreases the transfer characteristic slope. As a shunt diode stops
conducting, both the shunt resistance and the incremental transmission
increase in value.
For reference purposes, some commonly used dioderesistor combina
tions and their voltampere responses are tabulated in Fig. 227. The
first two diagrams illustrate a diode biased to start conducting, and then
one which stops conducting, as the voltage rises above the threshold
value. Figure 227c represents the response of a Zener diode, and Fig.
227d illustrates the control afforded over the breakpoint location by
placing two Zener diodes back to back. This final branch would normally
be used to insert biased break points into the series arm without having
recourse to actual bias batteries.
29. Dead Zone and Hysteresis. Two interesting examples of diode
wave shaping which are used in the analog solution of electrical, chemical,
and mechanical problems are the generation of the transfer characteristics
of dead zone and hysteresis. We shall first examine dead zone, which
appears whenever a threshold value must be exceeded before transmission
is possible. The steering system of an automobile has builtin dead
zone for safety; small disturbances of the wheel are not transmitted into a
turning movement. Some threshold value of steeringwheel rotation
must be exceeded before the automobile responds.
56
MODELS AND SHAPING
[Chap. 2
(a)
<b)
<e)
&
z**
r
■5*
Oi+0z
Ci+G z
B
r <h
(d)
■St Rz
M  I ft
Si Ez
fa
£2 «
Fig. 227. Various dioderesistor combinations and their voltampere response.
Fig. 228. Deadzone circuit and transfer characteristics.
Sec. 29] diode waveshaping techniques 57
The freezingandmelting process of pure materials presents a second
example of dead zone. Here the addition of heat produces a linear
temperature rise at temperatures below the critical point. But at the
melting and boiling points, a finite amount of heat must be added to a
constant temperature before there will be a change of state (solid to
liquid, liquid to gas). In the new state, again temperature increases
linearly with heat flow.
The simplest diode network which may be used to simulate a fixed
width dead zone employs, as the series branch, two Zener diodes connected
back to back (Fig. 228o). Their constant voltage drop in the Zener
region avoids the addition of any series bias source (a difficult problem
since this voltage must be isolated from ground).
We shall consider the response of the nonlinear series branch first.
On positive voltages conduction eventually becomes possible through
the forward diode of D x (D v ) and the Zener region of D 2 (D b ). How
ever, this will not occur until the branch voltage exceeds E,%. The
coordinates of the break point are
ei = E li . i = (223)
At zero current there will be no drop in R 2 and the input break coordi
nates are identical with those given in Eq. (223).
By symmetry, we can see that the other break point occurs at
ei = E.! i = (224)
Consequently, within the limits
E tl <ei< E zi
only the extremely limited transmission through the high inverse resist
ance of a physical diode is possible. Outside these limits the incremental
transfer slope is determined by the voltage divider composed of R x and
R2.
s = sHnr. < 2  25)
In the transfer characteristic of Fig. 2286 we note that the width
of the dead zone is a function of the individual Zener diode break voltages
and that the transmission slope is controlled by the series padding
resistor. The dead zone may be shifted to any output voltage by suitably
biasing the output resistor.
Hysteresis arises in many physical properties of materials, such as the
magnetization curve of iron and the stressstrain relationship. If the
forcing function is increased from zero to some final value and then
removed, the forced system parameter will remain fixed at a value other
58
MODELS AND SHAPING
[Chap. 2
than zero. But this is simply the object's memory which shows up as
the magnetic flux still present after the removal of the magnetizing cur
rent or as the residual stress in a steel rod after the stretching force is
released. Of course, limits are imposed by the physical nature of the
materials. Only so much flux can be supported regardless of the magni
tude of the magnetization current; this value is called the saturation
flux. In a stressed bar, excessive force will cause fracture.
To return a system (one which may be saturated but not fractured)
to its original state requires the application of a forcing function in the
opposite direction, usually of somewhat smaller magnitude than the
original signal. If the new input is too large, it will simply leave the
system in an excited state of the opposite magnetic polarity or with
the strain acting in the opposite direction (Fig. 2296).
*i
9 — VV\ —
W
D 2f
H
H °2
1
J2t e i2 D 3 y.
+ 1
E m =
¥
C±Z e 2
(a) (b)
Fig. 229. Hysteresis circuit and characteristics.
A smaller amplitude excitation will generate smaller area loops with
regard to both the displacement from the origin and the maximum output
amplitude. These are extremely difficult to simulate, and therefore we
shall concentrate our attention on the circuit of Fig. 229a, which gives
the response shown in Fig. 2296. The two diodes D 3 and Z> 4 limit the
maximum output excursion to ±E m and establish the saturation values
so necessary in many hysteresis systems. (Fracture may be simulated
by fuses.)
Operation is similar to that of the deadzone circuit. On a rising
input, the forward diode D lf and the Zener diode D iZ eventually conduct,
providing an output across C. Capacitor voltage then follows the input
to a maximum value determined by the clipping diode D 3 . As the input
falls, the charge on the capacitor (the memory of previous maximum
excitation) backbiases the formerly conducting diodes and, since no
discharge path exists, the output remains at the maximum value pre
viously reached. Eventually the falling input brings D xz and D if into
conduction, establishing a discharge path, and again the output follows
the input. The transfer characteristic now traces the lefthand portion
Sec. 210] diode waveshaping techniques 59
of the curve. If the input continues falling and then rises, the cycle
repeats, with the roles of the series diodes interchanged.
There is no necessity to drive the output into saturation. For genera
tion of hysteresis or backlash curves, smaller inputs generate the minor
loops indicated by the dashed lines of Fig. 229b.
The slowest input signal for proper response is determined by the
leakage of the capacitor charge through the diode inverse resistance, and
the fastest signal by the charging time constant
We choose C for reasonable response to the expected time range of the
input signals.
210. Summary. Many nonlinear systems whose externally measured
characteristics are available may be treated from the viewpoint of their
piecewiselinear representation. The nonlinear system, difficult to treat
in all its complexity, is reduced to a sequence of linear problems, each
with its own response and limits of operation. Simple mathematical
tools already at our disposal from studies of linear systems are sufficient
to reach acceptable engineering answers in those cases amenable to this
treatment. We can always handle circuits involving only dissipative
elements, no matter how nonlinear their characteristics may be. When
energystorage elements are also contained in the nonlinear system, the
problem becomes much more difficult.
If in each region the boundary conditions are independent (able to be
calculated only from the model and the boundaries of the region),
we can solve the system in a straightforward manner. If the constraints
involve past history, as in the hysteresis circuit, then when only a single
energystorage element is present, we are able to complete the analysis.
And if multiple modes of energy storage existed, the circuit would not
have yielded so readily to simple methods. Except in very special cases,
the problem faced is extremely difficult and beyond the scope of this text.
The first step in any analysis is the construction of the appropriate
models together with the clear delineation of their boundaries. Gener
ally, boundaries are determined by the energystorage elements and points
of conduction. Secondly, conditions of continuity across the region's
boundaries must be considered. These arise primarily from the original
characteristics and the type of energystorage elements involved, with,
however, the energystorage elements often determining the diode con
ditions. And finally we are ready to proceed to the solution.
The same techniques are used in solving many chemical, thermal, and
mechanical systems, where, for example, an ideal mechanical "diode"
might be postulated rather than an electrical one. A velocityoperated
clutch which engages or disengages at present shaft speeds is one mechan
60 MODELS AND SHAPING [CHAP. 2
ical example. Its slip on engagement is analogous with the diode for
ward resistance, while its imperfect disengagement represents reverse
resistance. In fluid flow, onoff pressure valves behave similarly, with
their pressure loss and leakage modifying the ideal valve in much the
same way as resistance modified the ideal diode. A metal plate, polished
on one side and black on the other, allows radiant heat to flow more read
ily in one direction than in the other; it may be used as our thermal diode
under the applicable conditions.
In many cases where it is difficult to formulate proper models, making
the grossest approximations leads to a circuit which can be solved. The
answer, admittedly incorrect, often gives some insight. This enables
us to refine the analysis later, eventually coming within an acceptable
engineering solution.
PROBLEMS
21. (a) By direct graphical construction evaluate the input voltampere charac
teristics of Fig. 230. Specify the individual slopes and the coordinates of the single
break point.
(6) Convert the circuit holding for each diode state into both Thevenin and Nor
ton equivalent circuits and find the break point from the two defining equations.
>2K
X lOmafrj
SI J
Fio. 230
22. Show how the transfer characteristics of Fig. 28 may be derived by a graphical
construction. (The voltampere characteristics of the input and of the shunt diode
branch must first be evaluated.)
23. A diode clipper such as the one shown in Fig. 27 is used in a system where the
input may vary from zero to a value that will never exceed four times the clipping
level. What series resistor should be chosen, expressed in terms of ry and r„ so that
the maximum error voltage on both sides of the clipping value will be equal?
24. A triangular wave such as the one shown in Fig. 215 (40 volts peak to peak)
is the input to the series clipper of Fig. 210. The bias battery is +5 volts, and the
diode parameters are r r = 100 K and r/ = 40 ohms. Sketch the transfer character
istics and draw the input and output waveshapes to scale when B\ equals (a) 25 K,
(6) 4 K, and (c) 1 K. In all cases calculate the peak values of the output signal.
26. Repeat Prob. 246 for the circuit of Fig. 29 where the +5volt bias is derived
from a tap on a 10,000ohm bleeder connected across a 50volt power supply.
26. We wish to clip the positive peaks of a signal at +10 volts by using a shunt
clipper similar to that shown in Fig. 27. The available diode has a forward resistance
of 100 ohms and an inverse resistance of 0.5 megohm, and in addition it has a Zener
break point at —25 volts.
DIODE WAVESHAPING TECHNIQUES
61
(a) Calculate the optimum series resistance and draw the circuit.
(6) Sketch the transfer characteristics, indicating the values of slopes and the
coordinates of all break points.
(c) Draw the output waveshape when the driving voltage is a 100volt peakto
peak sinusoidal signal.
(d) Under the conditions of part c, what angular percentage of the sine wave would
be transmitted relatively unaffected by the clipping circuit?
27. The single pulse shown excites the circuit of Fig. 231 at t = 0. If the initial
charge on C is —20 volts (polarity shown), evaluate the complete output response.
Specify all voltage and timeconstant values. Make all reasonable approximations.
100 K
+50 v
500
/(sec
Fio. 231
28. A 30volt peaktopeak 100Msecperiod square wave is clamped negatively at
+5 volts. The diode has a forward resistance of 50 ohms and an inverse resistance
of 100 K. We want to use the smallest possible capacitor so that this clamp will have
a fast response to changes in the inputsignal level, but we are absolutely restricted in
that the output must never exceed the clamping level by more than 2 per cent of the
peaktopeak input signal.
(a) What is the smallestsize capacitor which we may use? Round off your answer
to the larger even value, for example, 0.00228 == 0.0025. Explain why you would
use a larger value.
(6) How much energy must be restored in the capacitor over each cycle? (Make
any reasonable approximations to simplify your calculations.)
(c) If the peaktopeak signal is suddenly increased to 50 volts, how many periods
of the input signal will the circuit take to recover its clamping action?
(<Z) Repeat part c if the signal is decreased to 10 volts. (Hint: Consider the super
position of the squarewave input and the initial voltage across C. Find the time at
which the diode again conducts.)
29. The signal of Fig. 232 is the input to a clamp having r r = 1 megohm,
r, = 50 ohms, and C = 0.005 4.
(a) Sketch the output if this signal is clamped positively at zero and indicate all
important voltage values and time constants.
(6) Sketch the output if this signal is clamped negatively at zero.
(c) Under the conditions of part 6, how long after the termination of the large pulse
before the clamp capacitor is again charged? (See hint given in Prob. 28d.)
6 8 10 12 14 16 18 20 2
Fig. 232
t, msec
62
MODELS AND SHAPING
[Chap. 2
210. Solve the circuit of Fig. 215 for Ei, E' 2 , Ti, and Tt where r, = 10 ohms,
r r = 200 K, R, = 1 K, where the triangular wave period is 10 msec. Assume that C
is very large.
211. Consider the response of the circuit of Fig. 233 to the signal given. The
input has been at 200 volts for a very long time before the negative pulse appeared.
Draw the output waveshape, indicating all time constants and voltage values.
0.005 ^f
«1n,
I
ZOOv
~l
100
o
1
1
I
2 t, msec
Fig. 233
im:
+300
1 K<
212. Draw a circuit that will clip the positive peak of a 100ma peaktopeak tri
angular driving current at the 20ma point. Use a diode having a forward resistance
of 50 ohms and r, = 100 K and specify the optimum shunt resistance. Sketch the
output current, giving all important waveshapes. Explain how an otherthanzero
load resistance would affect the clipping action.
213. The current waveshape of Fig. 234 is to be clamped negatively at —5 ma.
The diode available has a forward resistance of 500 ohms and an inverse resistance of
1 megohm. The coil approximates an ideal inductance in that its resistance is only
10 ohms and its inductance is 50 mh.
(a) Draw the circuit used, showing where the bias current would be injected.
(6) Sketch the output current, indicating all important values and time constants.
(c) To what should the spacing between the pulses be changed so that the output
overshoot will just equal 1 per cent of the peaktopeak current?
(d) Repeat part 6 if a 500mh coil is used in place of the one specified above.
50
ma
t,
Msec
5
60 55
Fig. 234
100 105
214. What effect would the coil resistance Bl of a nonideal inductance have on the
current clamping action? Consider clamping a square wave of current at zero to
simplify your analysis.
215. Prove that the output waveshape of a current clamp will shift until the ratio
of areas is equal to the ratio of diode resistances.
216. A 20volt peaktopeak 1msec square wave is the input to the clamp of
Fig. 212. The diode has a forward resistance of 20 ohms and an inverse resistance
of 100 K; C = 0.01 rf.
(a) Sketch the output waveshape for the first two cycles if the square wave is
initially applied when it is zero, going positive.
(6) Repeat part o if a 0.001/if capacitor is shunting the diode. Make all reason
able approximations.
DIODE WAVESHAPING TECHNIQUES
63
217. Given a dc vacuumtube voltmeter with a 10megohm input impedance, our
problem is to construct an ac voltmeter. The dc unit reads correctly in conjunction
with a 1megohm probe. Two different ac probes are under consideration (Fig.
23Sa and b). In each case they will be connected directly to the voltmeter, i.e., not
through the dc probe.
0.05/»f
K
Input
5.5 M
VW
To
VTVM
Input
VTVM
(a)
(b)
Fig. 235
Calculate the VTVM reading for each probe for the various inputs listed below.
In each case relate the reading to some parameter of the input signal (rms, peak, or
average).
(a) 10voltpeak 1,000cps sine wave.
(6) 10voltpeak 1,000cps square wave.
(c) 20volt peaktopeak 1,000cps triangular wave having + 10volt average value.
(d) Signal same as in part c except that it has a — 10volt average value.
(e) 10volt 10/isecwide positive pulses with a period of 100 /isec.
(J) Repeat part e for negative pulses.
218. Plot the transfer and input voltampere characteristics of the circuit given
in Fig. 236. Sketch the output voltage if em = 100 sin at, giving all important values
(assume ideal diodes).
H
50 K
50 K
WW
:iook
25 K'
!5v
Fig. 236
Fig. 237
210. Draw the input voltampere and the transfer characteristics of the circuit of
Fig. 237 and sketch the output when a 50volt peaktopeak triangular wave is
impressed at the input.
220. (a) Set up the equations representing each region of operation given in
Table 23 (Sec. 28) and verify the values of all break points.
(6) Show the purely resistive model (TheVenin or Norton equivalent) implicit
within each region and specify the various component values. By direct solution of
the intersection of the models found, evaluate the break points bounding these regions.
221. Design a circuit which will have the transfer characteristics given in Fig. 238.
Specify all batteries and resistors, taking the smallest resistor as 10 K.
64
MODELS AND SHAPING
[Chap. 2
Fig. 238
222. In the circuit of Fig. 239, the input voltage is a symmetrical triangular wave
having a period of 16 sec, a peaktopeak amplitude of 80 volts, and an average value
of zero.
(a) Plot the output voltage as a function of time and label all break points with
their time and voltage values.
(6) The output signal is to be a rough approximation of a sine wave, with only the
break points agreeing exactly with the sinusoidal signal. What is the peak amplitude
of the sine wave we are approximating, and to what angles do the break points
correspond?
100 K
<?vW
eti
50 K
►HWVO+20V
50 K
H^VWo20v
._ 200 K tj ,200K
Fiq. 239
223. One method of multiplying two voltages together is to convert each voltage
signal into a current proportional to its logarithm (to the base 10) and then to add the
two currents in an arrangement similar to the summer of Chap. 1. If the output
voltage is kept small compared with the input, then we might employ a circuit whose
voltampere characteristics are logarithmic to reconvert log A + log B to the product
AB. This basic circuit is shown in blockdiagram form in Fig. 240.
ij«loge J
eo^me^
i 2 a\oge 2
I
Fig. 240
(a) As the first step, design a diode circuit which will approximate the logarithmic
voltampere response for positive input signals. Except for a scale factor, the input
current should exactly agree with log e ln at the following integer voltages: 10, 20, 40,
70, and 100 volts. Use 100 K as your smallest resistance value and specify all resistors
and bias sources.
DIODE WAVESHAPING TECHNIQUES
65
(6) If we use tables to reconvert the sum of the logs found from the two circuits of
part a, what error is introduced when multiplying the following numbers: 20 X 65,
20 X 50, 12 X 30?
(c) If we use a circuit identical with that developed in part a (except that the
smallest resistor is now 1,000 ohms) for reconversion of the sum of the logs to the
product of the input voltages, then
e„ ~ meiet
What is the circuit constant ml To what values would the errors of the products of
part b increase as a result of this additional circuit?
224. (a) Discuss the effects on the transfer characteristic of the circuit of Fig. 228
when we return the bottom of Us to a variable bias instead of directly to ground.
(6) A 100volt peaktopeak sinusoidal signal is applied at the input of this circuit.
Sketch the input and output to scale and determine the transmission angle if the fol
lowing circuit components are used :
lE,.
\E.,\ = 20 volts fii = 10 K R t = 100 K
225. Prove that the circuit of Fig. 241 also simulates a dead zone. Calculate the
transfer characteristic and specify all break points and slopes. What would happen
to the transfer characteristics if diode Z)i is transferred from point A to point B?
<f+ 50 v
►20 K
A]
1K<
1K<
1
2
w
:im
;20K
50 v +
Fig. 241
I
226. An 80volt peaktopeak symmetrical triangular wave is impressed at the
input of the hysteresis circuit of Fig. 229. The dead zone is symmetrically located
about the origin and has a width of + 10 volts. Sketch the input and output to scale
if the circuit saturates at ±30 volts. Ri is very small, and C is very large.
227. The circuit of Fig. 229 may also be employed as a memory, as may any
device exhibiting hysteresis. If the saturation region crosses the vertical axis, then
once the input drives the output into saturation, the output will remain there even
after the removal of the input pulse. Only by injecting a pulse of the opposite polarity
can the circuit be reset to the other saturation value.
(o) Design a circuit which will saturate at ± 10 volts and which will cross the «i =
line at this value. The only two available voltages are ±45 volts. Specify all
resistors and draw the transfer characteristic.
(6) Explain the disadvantages of setting the crossover point on the sloping side of
the transfer characteristic instead of in the center or near the center of the saturation
region.
(c) Sketch the output waveshape if the input pulses are those shown in Fig. 242.
66
MODELS AND SHAPING
[Chap. 2
+30 v
t
e
20 25
40 45 50 55
5 10 15
30 35
60 65 t, msec
30
Fig. 242
228. The temperaturevs.heatcontent curve of many pure Substances exhibits
one or more dead zones. These occur where heat of fusion or heat of vaporization
must be added to cause a change of state: solid to liquid or liquid to gaseous. For
example, the temperature of ice increases linearly with heat content from —20 to 0°C
and then remains there until sufficient heat has been added to melt all the ice. Tem
perature again increases with the addition of heat, but at a new rate to the boiling
point.
(a) We are interested in developing a circuit to represent this action where the input
voltage would correspond to the heat content (each volt to 2 cal) and with the output
voltage corresponding to the temperature (5°C/volt). Choose water at 0°C as the
origin of the axis and calculate the slopes, intercepts, and voltages needed to represent
a temperature range from —20 to 90°C. Design the diode circuit which will give the
required transfer characteristic. Try to use the minimum number of components.
(b) Repeat part o for a temperature range from —20 to 110°C. Choose a new
calorietemperature conversion ratio.
BIBLIOGRAPHY
Angelo, E. J., Jr.: "Electronic Circuits," McGrawHill Book Company, Inc., New
York, 1958.
Millman, J., and H. Taub: "Pulse and Digital Circuits," McGrawHill Book Com
pany, Inc., New York, 1956.
Stern, T. E. : Piecewiselinear Network Theory, MIT Research Lab. Electronics Tech.
Rept. 315, June 15, 1956.
Zimmermann, H. J., and S. J. Mason: "Electronic Circuit Theory," John Wiley &
Sons, Inc., New York, 1959.
CHAPTER 3
DIODE GATES
In contrast to Chap. 2, which treated diode networks excited at a
single terminal, this chapter will discuss circuits designed for operation
with multiple inputs. Where previously the complete circuit response
was characterized by the transmission and voltampere curves, this
simple representation now becomes quite inadequate. The multiplicity
of inputs interact, and consequently the state of any particular diode, at
any particular time, cannot be defined in terms of one input but rather
depends on the relative effects of the several sources. It is completely,
possible for a signal applied at one set of terminals to control the trans
mission path between another terminal and the output. Since it is
impossible to solve this problem in completely general terms, we shall
at least try to answer the following three questions in the course of the
analysis of specific circuits:
1. Does a transmission path exist from any one terminal to the output?
2. What conditions must be satisfied elsewhere in the network to
establish (or to interrupt) this path?
3. How does the circuit behave in the two regions separated by the
transmission threshold?
31. Application. Gates are circuits which make use of the bistate
diode properties for switching purposes. Defined, not in the narrow
sense of turning something on and off, even though this is included, but as
the establishment of information transmission paths upon the application
of a proper stimulus, diode gates find wide application in digital com
puters, control and measuring instruments, and stimulation systems.
In computers the gate output amplitude is usually of little or no impor
tance provided that it at least exceeds a preset threshold value. We
consider the output's presence as representing a "yes," or 1, answer, with
its absence as a "no," or 0, answer. Almost any desired information
for computing or control can be conveyed and operated upon by passing
a coded time sequence of pulses through various combinations of gates,
each pulse carrying one item of information.
In specialpurpose instruments, an external control signal, or proper
sequence of signals, opens and closes the gate, thus allowing or preventing
67
68
MODELS AND SHAPING
[Chap. 3
transmission of the information signal. Here the amplitude response
becomes important because this information may later be used for con
trol purposes.
Diode gates are classified, according to their performance, into three
groups : the or gate, which has an output when any one or all inputs are
present; the and gate, which allows an output only when all inputs are
applied; the controlled gate, where a controlling signal turns on the gate,
allowing transmission upon satisfaction of its other requirements.
Gate inputs are often short pulses. Consequently, to avoid introduc
ing reading errors, the gate should open fast to transmit the individual
pulse and close fast upon its removal to prevent false information from
appearing at the output. Any delay in opening and closing creates an
ambiguity in performance; therefore our investigation of the particular
circuits must be concerned with both their amplitude and time response.
t
«1
t
«2
1 1
1 1
! > '
1 1 '
1 1 1
t
1
II II
! ! ! !
t
1
*3
1
1
1
1
1
!
1 — 1
t
1
t
(a) (b)
Fig. 31. (a) Threediode or gate; (6) inputoutput relationships.
32. oe Gate. The circuit of a simple on gate designed for operation
with positive input pulses appears in Fig. 3lffl. For negative inputs
all diodes are reversed. The time relationship of the various inputs
to the output is illustrated in Fig. 316, ideal gate operation assumed.
For the initial discussion assume that the circuit is ideal; i.e., all source
impedances (r,i, r„2, and r, 3 ) are zero and all diodes switch between open
and short circuits. Suppose that at some instant ei > and ei = e 3 = 0,
the single diode D\ associated with e x conducts, establishing the trans
mission path between input and output. The output, now equal to e\,
is larger than d and e s , backbiasing their associated diodes. Only the
single source ei supplies power; it sees the load R. All other signal
sources work into the open circuits of nonconducting diodes.
When all inputs are greater than zero, the output is equal to the largest.
To prove this, apply the signals e x > e 2 > e 3 > and assume that
Sec. 32] diode gates 69
initially D 3 conducts. The resultant output e 3 is less than ei and e 2 ,
forwardbiasing their diodes. Thus there appears to be a transmission
path for each of the inputs. However, this argument is fallacious: once
Di conducts, the output reaches ei, the highest voltage of all applied
signals, backbiasing the diodes D 2 and D 3 . Since we can only maintain
the single transmission path through Di, the output of the or gate auto
matically becomes equal to the largest input signal at each instant of time.
The or gate may be considered analogous to a group of normally
open parallel relays or switches connected from various signal sources
to the common output. Closing any one provides an output regardless
of the state of any of the others. A mechanical system of several power
sources coupled through clutches to the same drive shaft behaves sim
ilarly. The engagement of any single clutch furnishes rotational power,
and if ratchet or slip clutches are used, only the one connected to the fast
estrotating power source transmits power. All others slip (backbiased).
Often a number of instrument or telemeter outputs whose source
impedance may be too large to be ignored are the signals driving the gate.
Before we can begin calculating the corrected amplitude response, we
would like to know each diode's state so that it can be replaced by
either an open or a short circuit. If only one input is applied, this
problem is trivial. But suppose that all inputs are simultaneously
excited by voltages of the same order of magnitude; how do we now find
each diode's state? We might, as a first attempt, guess that they all
conduct. Then, by superposition from the model found by replacing all
diodes by short circuits (r, 3> r t ) , the output is
„ „ «r.,r., m R 11 r.i  r. z , R \\ r.i H r„
r.x + R  r. 2  r„ B1 ^ r„ + R  r.x  r., "' ' r„ + B  r fl  r„
(31)
The next step is to compare e„ found above with each input (ei, e 2 , e 3 ).
If it is larger than one or more, then the diodes connecting the smaller
amplitude generators are backbiased and those particular inputs cannot
contribute to the output. For example, when e„ < ei, e„ < e 2 but e„ > e 3 ,
we must modify Eq. (31) by opening the transmission path from e%
to the output. The output now becomes
J _ Br rt ei+ gK» 62 (3.2)
~ r.i + R  r.^ 1 ^ r.. + R \\ r.,
(Jnder the circuit conditions leading to Eq. (32), signal sources ei
and e 2 are loaded by both R and each other; e% isolated by its backbiased
diode need not be considered. The signal power requirements are deter
mined by the current each source must supply. This depends upon the
difference between the opencircuit terminal voltage and the output
70
MODELS AND SHAPING
[Chap. 3
voltage, and thus upon the contributions from all other signals,
example, the current flow from e x is
Ci — e
For
»i =
r.i
If the or gate contains more than the three sources shown, the addi
tional ones are treated similarly, contributing to the output only when
their particular diode conducts.
Transient Response. Our investigation of the transient response of the
or gate will initially concern itself, for simplicity, with the twodiode
gate shown in Fig. 32. High source impedance degenerates the pulse
waveshape by increasing the circuit time constants; if we are interested
in fast response, the gate must be driven from such lowimpedance sources
as cathode followers or transformers.
f
H
>r
— Wv
H —
r r
WV
VvV
Fig. 32. Twodiode or gate — piecewiselinear circuit.
In the above circuit the diode shunt capacity, only about 1 or 2 /x/if ,
is obviously much less than the total stray output capacity (10 to 50 wi).
By neglecting the small term, we avoid the necessity of calculating the
response of a circuit containing two independent energystorage elements,
the series combination of r.j and C 2 in parallel with R and C . Further
more, a gate should not attenuate the input pulse excessively while
transmitting it. This requires that (r,  R) » (r f + r.i). Under these
conditions the output's peak value will be almost the same as that of the
single input, rising toward this peak with a time constant t\, which is
found from inspection of the simplified model.
ri = [(r.i + r,)  R  r r ]C. S (r.i + r f )C, (33)
After the input pulse disappears, the output decays toward zero with
the much longer time constant
(34)
 = («U r i)'
Sec. 32]
DIODE GATES
71
Figure 33 shows the applied input and the resultant output pulse.
We might observe that for an extremely narrow input pulse, e might
never reach the final value and might not even pass the arbitrarily
designated threshold before the input drops to zero. In addition, the
long decay time constant smears the trailing edge, causing the output
to persist for some time after the input's removal, a false indication that
the input is still present. We see that the gate's resolution is severely
limited by its time constants; it cannot distinguish extremely narrow
pulses.
t
t
e
r
\
Threshold
*
I
s.
l«At^ t ' kAtH t
Fig. 33. ORgate input and resultant output pulse.
I — VWA/W
Fig. 34. ndiodegate equivalent circuit — singlediode excitation.
Computer or gates, receiving information pulses from many sources,
usually contain appreciably more than the two diodes shown in Fig. 32.
We suspect, even before we begin calculating the response, that the
additional ones will affect the pulse waveshape through the increased
resistive and capacitive loading introduced. In any or gate, when only
one diode is excited, all the others are backbiased and in parallel.
This is indicated in the equivalent circuit of an ndiode or gate (Fig. 34).
Furthermore, it might reasonably be expected that, in any one system,
the inputs would be supplied from similar sources to nearly identical
diodes. Since all the backbiased diode branches are identical, the same
voltage must be developed across each branch element, thus allowing us
to connect equivalent elements together (the dashed line of Fig. 34).
By doing this we parallel all source impedance (r,i, r.j, . . . , r,„);
and as n increases, this combination decreases proportionally, approach
ing zero as a lower limit. Even though it is difficult to justify the
72 MODELS AND SHAPING [CHAP. 3
omission of a single source impedance, it is relatively easy to justify the
neglect of the verylowresistance parallel combination. The reduoed
circuit places all capacity in parallel, becoming a singleenergystorage
element system. Furthermore, R  [r r /(n — 1)] » (r.i + r f ), prevent
ing excessive pulse attenuation by the gate. As a consequence,
n £g (r.i + r f )[C t + C d {n  1)] (35)
where C d is the shunt capacity of the individual diode. The increase
in time constant, completely due to the diode capacity loading, slows
the initial rate of pulse rise. Therefore, if narrow pulses are to be trans
mitted, we must limit the number of diodes in any one gate.
After the termination of the pulse, the energy stored in the output
capacity backbiases all diodes and the output decays with the slow time
constant
r 2 £ R  J (Co + nCi) (36)
For n sufficiently large, the circuit resistance during decay is primarily
determined by the reverse resistance of the parallel diode. It follows
that Eq. (36) may be rewritten
nSC.J +C d r r (37)
The first term of Eq. (37) is obviously much less than the twodiode
decay time constant [Eq. (34)], and it further decreases with an increas
ing number of inputs. The second term is the very small recovery time
constant of the individual diode. We come to the surprising conclusion
that the additional diodes may even improve recovery through reduction
of the circuit time constant. This is the direct consequence of the circuit
resistance decreasing faster than the capacity increases. However, the
greater loading of signal source has the adverse effect of increasing both
the attenuation of the pulse and the source power requirements.
33. and Gate. The and gate determines coincidence, presenting an
output when, and only when, all inputs are simultaneously excited. Its
operation may be compared to a parallel combination of normally closed
switches shorting the output to ground; only after opening all switches
will the output be other than zero. In hydraulics, the analogous circuit
is a series of check valves, all of which must be opened before any fluid
flows from the source to the sink.
A simple diode and gate operating on positive pulses appears in Fig.
35a, with the inputoutput time relationships shown in 356. Since
transmission of negative pulses is possible without coincidence of inputs,
we include at the output a diode D whose function is to prevent negative
Sec. 33]
DIODE GATES
73
all diodes and bias
excursions. For operation with negative inputs,
voltages are reversed.
When ei and/or e 2 are zero, at least one diode conducts, shorting the
output to ground. (Ideal diodes and zero source impedance, as shown in
Fig. 35o, are assumed for this initial discussion.) However, if signals
44
D 2
M
!
e 2
t
e
t
l_
1
t
*■
t
(a) (»
Fig. 35. (o) Ideal and gate; (6) ANDgate inputoutput relationships.
are applied so that Eu> > ei > e 2 > 0, only diode D 2 conducts, and it
connects the output terminal directly to the source e 2 . Diode D\ is
backbiased, removing ei from the circuit. The output, consequently,
will always equal the smaller of the ivputs. A second possibility arises
where both input signals are greater than Em,, backbiasing both diodes.
In this case the output rises to En, its
maximum possible value.
Transient Response. Having es
tablished the basic behavior of the
and gate, we are ready to proceed
to the quantitative analysis of the
possible modes of operation defined
above, using, for this, the complete
model of Fig. 36.
Case 1: E» > ei > e 2 >
Fig. 36. Model of an and gate showing
parasitic capacity and resistance.
Here the output rises to approxi
mately e 2 from an initial voltage
slightly above zero (due to the drop
across r, + r/). Exact initial and
final values are easily found by drawing and solving the resistive models
holding. Before the pulses are applied, both diodes are conducting, but
after reaching steady state, only D 2 conducts. Even a superficial exam
ination convinces us that the condition necessary to minimize the small
74 MODELS AND SHAPING [CHAP. 3
initial value and simultaneously to maximize the final value is
r. 2 + r, « R « j (38)
Since the stray capacity prevents any instantaneous change at the
output, upon injecting the input pulses all diodes are immediately driven
off. The output rises toward En, with the time constant
n = ( R I! ^j (C. + ZC d ) (39)
However, it will never reach Ebb) before it can do so the output will
equal the smaller of the two inputs, e 2 . D 2 is again forwardbiased,
and it limits the rise. At this point the charging suddenly ceases. The
time required for the complete rise is given by
h = ti In Ebb (310)
atb — e 2
When the smallest pulse is 30 per cent less than En,, this time is only
1.2ti. Limiting e 2 to an even smaller percentage of Eu, will further reduce
the circuit response time : with a pulse of 0.5En, the rise time is appreci
ably less than one time constant; it is only 0.69n.
As soon as a single input returns to zero, the connecting diode again
conducts and, using the inequality of Eq. (38), the output decays with
the time constant
r 2 = (r„ + r,){C, + 20,) (311)
Obviously, the decay through the on diode is much faster than the initial
rise.
Case 2: ei > e 2 > Em,
Under these circumstances the output will always be less than the
amplitude of the applied excitation. The coupling diodes are initially
driven off and remain in this state until the input pulses disappear.
There will be no shortening of the output rise, which, within approxi
mately 4ri, reaches Em,. Even if the rise time is defined as in Eq. (132)
(the time for the output to rise from 10 to 90 per cent of Ebb), it will still
take h = 2.2ti sec to reach an acceptable final value.
The decay in this case is identical with that found for case 1.
A graphical cbmparison of these cases, where the value of Ebb is adjusted
so that both outputs are of the same amplitude, appears in Fig. 37.
We conclude, from this sketch and from the relative rise times calculated
above, that case 1 is the preferred class of operation, especially if we can set
Ebb well above the pulse amplitude. It also offers other advantages
Sec. 34]
DIODE GATES
75
in that the output pulse is not clipped during transmission and that it is
available at low impedance (through the conducting diode) for direct
application at the input of the next gate.
Additional input diodes change the gate response primarily by increas
ing the stray capacity and reducing the effective reverse resistance
during the initial pulse rise. Arguments similar to those used with the
or gate lead to the same general conclusion: when extremely narrow
pulses are the gating signals, the number of diodes in the gate must be
limited; otherwise the gate will distort the pulse beyond all. recognition.
E t
Co
/
/
1
1
e 2 1—;
t,
Case 1
Fig. 37. Comparison of the output pulses under the two modes of ANDgate operation.
t
C 2 * D 2
Cc ^ At
", innn
in
— »»
t
n
n
n
t
n
n
t
Fig. 38. Controlled gate, andor, and the applied inputs and resultant output.
34. Controlled Gates. Almost any method of either shifting the
dc operating point of the gate or applying the control voltage in such a
manner that it prevents transmission converts an ordinary and or or
gate into a controlled one. An extremely simple change results in
the circuit of Fig. 38, where the pulses are accoupled rather than
dccoupled as previously. Changing the input circuits back to the
direct coupling used in Sees. 32 and 33 will not change the basic opera
tion; it only changes the required driving signal.
76 MODELS AND SHAPING [CHAP. 3
The two input diodes constitute a simple or gate transmitting to
point b any positive pulses applied at e x and/or e 2 . However, if D is
kept conducting, the output pulse will have to be developed across its
very low forward resistance. Consequently, the output remains close
to zero regardless of the signals ei and e 2 . The additional control diode
D c and its very negative bias voltage E ce serve to maintain this condition
by establishing a large forwardcurrent flow. Diode D also performs
the second function of preventing negative excursions at the output.
A large positive control pulse applied at e c backbiases the previously
conducting control diode and keeps it backbiased until C c can recharge
through R e . This pulse effectively disconnects the bias voltage from the
output, allowing transmission of any positive pulses now applied at
e\ and/ or e 2 . We see, therefore, that the condition for opening this gate
is the simultaneous excitation at e c and e\ or e 2 .
Unless the inputcircuit time constants are very long in comparison
with the pulse duration, they will distort the applied signals, leading to a
correspondingly poor output waveshape.
Returning the control diode of Fig. 38 to a positive bias voltage En,
rather than the negative one shown, changes this gate into a notor
circuit. The positive voltage nor
\ bb mally backbiases D c so that it
plays no role when pulses are
applied at ei and/or e 2 . But if
simultaneously we apply a larger
i t * h? . negative pulse at e c , it establishes
a conduction path through D„ and
, D c , maintaining the output at zero,
independent of ei and e 2 . This
negative pulse is sometimes referred
Fig. 39. Threshold gate. to as an inhibiting signal since its
function is to prevent the gate
from opening. A similar modification of an and gate produces an
equivalent notand circuit.
Another type of controlled gate appears in Fig. 39, where we can
recognize the two input diodes as an and gate. T)\ and D 2 normally
conduct, maintaining the voltage of point a at zero, provided that at
least one input remains zero. C charges to the negative control voltage
e c . To prevent this negative voltage also appearing at the output, we
decouple through the diode D .
After opening the and gate (large positive pulses at all inputs), what
ever signal appears at point a must be transmitted by C to point b. If
e c is a slightly negative dc bias, D„ will now be forwardbiased, allowing
the output to rise to the difference between the minimum applied input
ei° K
D 2
e 2 ° K
Sec. 34]
DIODE GATES
77
pulse and the control voltage. When ei is that particular pulse, the out
put becomes
(312)
ei
e c >0
where e c , the initial charge on the coupling capacitor, may be considered
as the transmission threshold. Coincidences of input signals of greater
amplitude are transmitted, while those of lesser amplitude cannot establish
a transmission path and will not produce any output [Eq. (312)]. If
the threshold is set slightly above the noise level, then the random noise
pulses will not be transmitted and will not register falsely at the output.
f
12345678
nnnnnnnn
nnnnn
t
/
/
\
\
'c
1
1
X
N.
/inrih
4 5 6 7
Fig. 310. Controlledgate inputoutput pulses.
An alternative mode of gate operation occurs when we initially set e c
negative enough to prevent transmission for all possible amplitude input
pulses. Only after a positive control signal e c charges the capacitor
to a voltage close to zero will satisfaction of the and gate produce an
output. This switching action is similar to that discussed for the first
controlled gate considered. But where the first gate was primed immedi
ately upon the injection of the control pulse, this gate is not ready for
transmission until considerably later, until C charges. Even though
one of the input diodes always conducts, the charge and discharge time
constants R C C are quite long. In the pulsetime relationships (Fig. 310) ,
note that the direct consequence of this is very poor transmission of
pulse 4, good transmission of pulse 5, proper transmission of pulse 6, and
false transmission of pulse 7. Obviously, use of this gate must be limited
to pulse trains having wide separation. Replacing B c with a diode con
nected for fast charging of C during the positive control pulse improves
the turnon time, but not the gate's final recovery time.
78
MODELS AND SHAPING
[Chap. 3
35. Diode Arrays. One problem which arises quite often when
information must be transmitted along several channels is how to select
the proper one out of a number of possibilities. But the problem may
be even more complicated; the same input may have to be transferred
between various paths in a controlled time sequence. We can do
this by using a number of independent diode gates. One input to each
gate is used for the information to be transmitted while the other
inputs are used for the control signals, i.e., the pulses that open and
close the gate at the appropriate times.
Control pulses
bed
«\
k.
X
X
X
X
X
k.
X
o6
Fig. 311. and array used for transmission selection.
.*^.
X
°1
°2
Outputs
°4
X
°5
X
When these gates are combined into one matrix, we refer to the
resultant circuit as a diode array. Figure 311 illustrates one such array
where the excitation of two out of four control channels opens an infor
mation path for negative signals. The diodes connected to any one of the
information channels (1 to 6) constitute an and gate. As they are
normally conducting through the controlpulse generator, they establish a
lowimpedance path, shunting the signal to ground. However, if large
negative pulses are simultaneously applied to two out of the four control
channels (a, 6, c, and d), the associated pair of diodes are backbiased,
the gate is opened, and transmission along one path becomes possible.
Sec. 36] diode gates 79
For example, the excitation of a and b opens path 1, the excitation
of a and c opens path 2, etc.
The statement of the particular twooutoffour coding used for the
selection of one out of six channels may be presented in a tabular form,
called a truth table. We indicate by the symbol 1 that a pulse must be
present in a particular control channel and by the symbol that it may be
absent. The array of Fig. 311 satisfies the following statement:
Pulse
inputs
Channel
a
b
c
d
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
If all signal sources in the various channels (1,2,..., 6) are identical,
e.g., a constant negative voltage E cc , then upon the application of the
appropriate control signals, an output pulse will appear in a single
channel. Thus the array translates the two pulses ab to mean number 1,
as shown in the truth table. The simultaneous excitation of two out of
four inputs steers the output to one point out of the six possibilities.
More complex logic situations may be satisfied by adding additional
diodes in a larger matrix. These may appear as ors as well as ands,
and if both positive and negative control pulses are available, not
diodes can also be included.
36. Diode Bridge— Steadystate Response. Under circumstances
where the amplitude of the output contains important information, the
gate requirements become extremely critical. Neither of the circuits
examined in Sec. 34 is satisfactory for amplitude gating; in both the
output voltage not only varies as a function of the control signal, but
also contains some percentage of the control signal, even when all inputs
are zero. For example, in the gate of Fig. 38 a small negative voltage
is developed across the forward resistance of the output diode as a direct
consequence of the current flow to E cc .
One convenient method of avoiding these errors is by using a balanced
system driven by two equal signals of opposite polarity. The error
voltage produced by each signal will be equal and opposite, canceling out.
Since we know that in linear circuits almost complete voltage cancella
tion takes place in a balanced bridge, wj might well consider the appli
80
MODELS AND SHAPING
[Chap. 3
cation of a diode bridge for amplitude gating (Fig. 312). For the output
to be completely free of the control signal, i.e., not contain any component
directly contributed by this signal, the bridge must remain balanced
under all conditions. Or rather, we must limit circuit operation to the
conditions that will not unbalance the bridge.
Two balance conditions are obvious, the first when all diodes conduct :
111
(313)
and the second when all are backbiased :
Tri
(314)
To satisfy both Eqs. (313) and (314), the same type of diode is not
only used for each arm of the bridge but is even individually selected and
matched so that all will have identical forward and reverse resistance.
Sometimes to further improve the
balance, additional series and shunt
resistors are added to each diode.
Other conditions of bridge balance
are also possible, for example, D x
and Z) 2 on and D 3 and Z>4 off or vice
versa. We shall see below that this
represents an impossible mode of
bridge behavior and need not be con
sidered. The remaining possible
combinations of diode states occur
when one set of diagonal diodes, Di
and Dt,, conducts and the other set,
D 2 and D 3 , is backbiased. But this leaves us with an unbalanced bridge,
an undesirable situation which must be avoided.
From the above discussion, we conclude that the sole function of the
control or gating signal is the closing and opening of the gate, i.e., bring
ing the diodes into and out of conduction. Any large enough signal
will cause switching, but to guarantee definitive diode states the signal
should switch them smartly. For this reason we choose a symmetrical
square wave, having a peaktopeak amplitude of IE for our gating volt
age. If this signal is supplied from a source unbalanced with respect to
the balance point of the bridge (in this case ground), then all diodes will
not be equally excited, and with a very unbalanced drive some diodes may
remain completely unexcited. We must therefore either use a balanced
Fig. 312. Diode bridge.
Sec. 36] diode gates , 81
generator output or drive the bridge through a transformer. For the
polarity indicated in Fig. 312, all diodes are conducting, and on reversed
polarity, all become backbiased. However, this statement implicitly
assumes that the contribution from e\, which will be examined below, is
not large enough to turn off a conducting diode or to turn on a non
conducting one.
When e„ forces all diodes into their active region, they present a very
low resistance from the output to ground. By replacing them with their
forward resistance, we see that across
either diagonal ac or bd (Fig. 312),
the resistance is the same as that
of a single diode r f . Secondly, when
the bridge is properly balanced, the
voltage from 6 to d, due to e„, must be
zero; it depends only on e\. Like
wise, the voltage developed from a
to c depends only on e c . ?™ 3 : f 13 . Equjyatart bridge circuit
J the switch position is a function of the
The reversal of the control signal control signal,
backbiases all diodes, allowing us
to replace them by their reverse resistance. The resistance from output
to ground, bd, now becomes r r , and if the bridge is still properly balanced,
again e c will not contribute to the output.
Thus the gating signal causes sequential shunting of the output, first
by r f and then by r r . The circuit related to ei appears similar to the
shunt clipper of Chap. 2 where the clipping diode shunted the output
by its forward resistance to prevent undesired transmission and by its
reverse resistance to allow desired signal transmission. This leads us
to interpret bridge operation as controlled clipping with its information
transmission path a function of the control signal period (and voltage)
rather than input signal amplitude. Since the two circuit states shown in
Fig. 313 are the same as those existing in the clipper, the problem
posed in solving for the optimum value of Ri must lead to the identical
value found in Chap. 2 [Eq. (29)] :
Ri = y/r T r f
We shall now direct our attention to the input signal and examine the
circumstances under which it could unbalance the bridge by changing the
state of one or more diodes. A straightforward and convenient approach
is to first calculate, and then compare, the two current components in
the individual diode, one produced by the gating source and the other
by the input signal. When the total diode current changes sign, the
diode changes state. The individual diode current component maintain
ing the bridge closed (diodes conducting) is supplied by the gating signal.
82 , MODELS AND SHAPING
Solving the model of Fig. 314 yields
E
lea —
2(i2. + r/)
[Chap. 3
(315)
The maximum contribution from e\ occurs at the peak of the input
voltage.
El m
/l» =
2(Ki + r,)
(316)
The total current supplied by each signal is twice the current flow in the
individual diode due to that source.
6+ E 6
Fig. 314. Bridge current flow during conduction (all bridge resistors are r/).
In Fig. 314 the direction of the individual current components is
indicated by the arrows. Note that the two components subtract in Di
and Di but add in D s and D s . The former diode pair is in danger of
being forced out of conduction on the positive input peaks, and the latter
on the negative peaks. To prevent this, the contribution from the gating
signal [Eq. (315)] must always be greater than the current supplied by
the signal source [Eq. (316)]. By writing the required inequality, we
arrive at one relationship between the three remaining unknowns E,
E lm , and R.:
Ely,
Ri
i>
(317)
Equation (317) takes the form given since both Ri and R. are much
greater than r t .
For a second relationship, we naturally turn to the second bridge
state, the open gate (all diodes backbiased). The individual diode
current supplied by the gating voltage flows in the direction opposite
Sec. 36] diode gates
to the previously found /„, and its amplitude is
/c» =
E
2(B. + r r )
83
(318)
The current furnished from the signal source still can flow in the same
direction as I^>, but its value changes to
Ev
2(Ri + r T )
(319)
Under these circumstances the direction of current flow is such that on
the positive inputs the two components add in Di and Z>4 and subtract in
Di and Dz. Conditions established may drive one pair of diagonal diodes
into conduction on the positive peak and the other pair on the negative
peak of the input signal. Of course, this would again seriously unbalance
the bridge, and to prevent this, we must also satisfy the second inequality
E h
E
Ri + rr R. + rv
But since R, 4C r P and Ri <K r r , this simply reduces to
E lm < E (320)
An additional complication arises because a diode's resistance is not
constant in each region as depicted by the piecewiselinear model, but
varies rather widely with its cur
rent. By taking the slope of the
curve (Fig. 315), we see that
the forward incremental resistance
decreases from a relatively high
value at small currents to a very
low nearly constant resistance for
large forwardcurrent flow. The
inverse incremental resistance in
creases with increasing reversecur
rent flow to some maximum, and
then begins decreasing.
100 ma ■ ■
SO ma
100 v 50 v
100 jta
12345»
«d
Fig. 315. Diode characteristic.
Whether the bridge is open or closed, the additional current contri
bution from the signal source increases current flow in one and decreases
it in the opposing diagonal diode pair, thus decreasing the resistance of
the first and increasing that of the second set of diodes. These opposing
changes just compound the unbalancing of the bridge. We can minimize
this secondorder effect through setting the quiescent point at the center
of the forward linear region of the diode and permitting only small current
excursions about this point. Since E/2 appears across each backbiased
84
MODELS AND SHAPING
[Chap. 3
diode (R, <K r r ), this voltage should place the diode in the center of its
reverse linear region for optimum bridge performance.
Examination of Eqs. (315), (316), (318), and (319) indicates that
in both regions, e c may be considered as setting the quiescent current
and ei as causing the incremental current variations about this point.
And as in all piecewiselinear circuits, when the peak incremental current
exceeds the quiescent value, the circuit limits. We conclude from Eq.
(315) that with any given peak driving signal E, we should choose R,
to supply sufficient forward current to set the conducting bridge in its
linear region. In addition, in order to maintain smallsignal performance,
E\ m must be limited to some small fraction of E [Eq. (320)]. The exact
t
nfintN
mr^
Fig. 316. Bridge inputoutput signal relationships.
percentage depends on the secondorder variation of diode resistance and,
of course, will vary with diode type. Generally, satisfactory results
have been obtained on limiting the signal to 10 to 50 per cent of E.
Waveshapes of the input (squarewave control signal superimposed)
and the resultant gate output, illustrating the gate's operation, are shown
in Fig. 316. Note that the output appears only during the negative half
period of the gating signal when the bridge is nonconducting. The gating
signal chops the input into a train of equally spaced varyingamplitude
pulses. For this reason the bridge gate is often referred to as a chopper,
or a bridge modulator.
Many alternative circuit configurations are practicable, each solving
a particular amplitude gating problem. The bridge can be interposed in
series with the transmission path rather than in shunt. It can be used in
DIODE GATES
85
ei
3
^S' 1
3
ixZj
o
o
s
o
o
o
§
o
Fig. 317. Bilateral transmission bridge.
Sec. 37]
a pushpull system or the singleended one previously discussed, driven
by direct current pulses, or even sinusoidal signals. The bridge might
be used for bilateral as well as unilateral transmission, since its resistance
is independent of signal direction.
One feasible circuit, .illustrating
many versatile attributes of the diode
bridge, is shown here in Fig. 317.
The bilateral transmission path (ei to
c 2 and vice versa) is instituted by
forwardbiasing the diodes and inter
rupted by backbiasing them.
37. Diode Bridge — Transient Response. The response of the diode
bridge to the leading and trailing edges of the control signal determines
the switching time and, consequently, the output waveshape. Moreover,
because of the interaction within the gate, the input signal will likewise
play a role in determining the response of the bridge to the gating signal.
An exact analysis of the total problem becomes unreasonably tedious, but
the approximate response, sufficient to delineate the limitations of the
bridge, is not difficult to calculate. We divide the problem into two
parts : first, we find the time required to turn the bridge on and off with the
signal set at zero; second, we calculate the response of the bridge output
with respect to the input, while the bridge alternately conducts and
opens.
Fig. 318. Bridge model for the calculation of turnon, turnoff times.
For calculation of the turnon, turnoff time, we can use the model
of Fig. 318 and, if the bridge is perfectly balanced, the parallel com
bination of Ri and C„ will not influence its response. Assume that the
bridge is open (diodes off), on the verge of switching closed. Since
86 MODELS AND SHAPING [CHAP. 3
R. «: r r , the initial charge on each diode capacitor is —E/2, maintaining
the diode cut off.
At t = 0, e c changes polarity. Each capacitor starts charging toward
the new steadystate voltage of +E/2 with a time constant found from
the seriesparallel diode branches,
n = (r,  R.)C d ^ fi.C« (321)
In calculating Eq. (321) we might note that, in terms of the effective
resistance and capacity, the four diodes in the balanced bridge look
like a single diode. Thus the exponential response of the voltage across
a single diode becomes
e d = yiE  Eer'i* (322)
Eventually this voltage reaches zero and the diodes change state — they
begin conducting. Equation (322), evaluated at e<» = 0, gives the bridge
turnoff time as
t x = ti In 2 (323)
The nowactive diodes reduce the circuit time constant to the almost
insignificant t 2 .
Ti S C d rf
Cd continues charging, but toward a new steadystate value E it found
by taking the Thevenin equivalent voltage across each diode in the model
holding for this region.
When the control signal again changes polarity, the energy stored
in the shunting capacity maintains conduction for some short time.
Instead of initially charging toward —E/2, each diode charges toward
E *=M. E
with the time constant T2.
The bridge turnon time (diodes cut off) becomes
tt = r 2 In 2 (324)
which is the same fraction of the appropriate time constant rt as the
turnon time was of ti [compare Eqs. (323) and (324)]. On reaching
zero after this very short time, the diode turns off and its capacity
recharges to —E/2, with the reverse time constant n. Recovery is
virtually complete within four time constants. Figure 319 illustrates
Sec. 37]
DIODE GATES
87
the behavior of a single diode, showing both its turnon and turnoff
times.
When the gating period is sufficiently long, these switching times
become insignificant. They do, however, set a definite lower limit to the
usable squarewave period. In addition, any capacitive unbalance causes
unequal diode turnon and turnoff times, unbalancing the bridge and
producing sharp exponential pulses at the output. Sometimes additional
trimmer capacitors are placed across each diode and adjusted for proper
capacitive balance. This, of course, increases C<j, and consequently the
time required to switch the diodes from on to off, and vice versa.
Fig. 319. Singlediode voltage waveshape produced by e*
The solution of the second part of the response problem, i.e., calculation
of the output waveshape with respect to the input itself, becomes almost
trivial once we draw the proper model. Since the bridge behaves
as a controlled clipper, it might well be represented as two resistors
and a switch controlled by e c , alternately connecting one and then the
other (Fig. 320a). The circuit has two time constants: a long one for
the rise t» and a short one for the decay T4.
t, S «,(C. + &) r 4 S* r,«7. + C d )
(325)
We can expect t% to be much longer than the bridge turnoff time con
stant r\ [Eq. (321)], as a consequence of the extra stray capacity at the
output, and it will determine the fastest allowable gating signal.
If we now attempt to find the actual bridge response by combining the
response of each signal taken individually, we find ourselves facing a
rather difficult problem. The calculation of turnon and turnoff times
had ignored the presence of the input signal, which will, during the turn
off interval, contribute additional current to one diagonal pair and
reduce that flowing in the other pair. A rising voltage appears across the
diodes because of the gating signal (Fig. 319) ; therefore Eq. (320) will
not remain satisfied until the gate is completely off. The two diagonal
88
MODELS AND SHAPING
[Chap. 3
diodes D 2 and D% conduct first, unbalancing the bridge. A similar effect
appears during the turnon time; one pair of diagonal diodes cut off
before the other pair, again causing bridge unbalance.
Because of this unbalancing, exponential overshoots contributed by
the gating signal are evidenced at the output (Fig. 320c). By recon
sidering the previous simplified discussion, the method of minimizing then
becomes apparent. When the time constants n and t 2 are small, the
bridge recovers fast, remaining unbalanced for the smallest possible time
and producing only narrow overshoots. Excessive output capacity
may also slow circuit response, preventing the rapid rise shown below.
Ri
9— MV
f(ec)
4*2
\ f\ r
(a)
(b)
Fig. 320. (a) Bridge inputoutput equivalent model; (b) ideal output waveshape
(constant input voltage) ; (c) actual output response.
However, these overshoots do improve the time response of the bridge,
and if they are sufficiently narrow they may not disturb us at all.
38. Concluding Remarks on Diode Gates. Most gates may be
broken down into combinations of the various types discussed in this
chapter. The methods of analysis used are applicable to all since the
problems faced are similar: rise and decay times, steadystate response,
and the time required to open and close the gate.
Our discussion of the and and or gates purposely omitted the last
point in order to simplify the first look at the subject. The student may
now go back and with the material from Sec. 37 reexamine the earlier
gates. In those gates the energystorage elements across the diodes also
prevent the immediate change of state; they maintain the previous condi
Sec. 38] diode gates 89
tion in the face of circuit or excitation changes until they charge or
discharge, as the case may be.
In the first sections of this chapter, the time constants given are
those which affect the response after the diode has changed state. With
diodes in common use, the opening and closure times are too small to
be significant except when we try gating extremely narrow pulses. Of
course, any additional stray capacity will change the circuit behavior and
increase the significance of the switching delay.
Semiconductor diodes introduce additional delays in that they them
selves do not immediately switch from on to off and back again upon
pulse excitation. As a consequence of their physical structure, they
have three inherent delays:
1. Turnon time
2. Turnoff time
3. Storage time
Turnon and turnoff times correspond to the very short time required
to inject and sweep out the majority charge carriers in the diode materials.
They depend on the physical structure, diode material, junction area,
and mean electrical path. Generally, these times are relatively insig
nificant in determining total response, as they are much less than the
storage time.
Some majority carriers are transported across the diode junction into
the other material, where they become minority carriers; i.e., the current
flow carries holes from the p to the n material and electrons from the
n to the p material. Upon suddenly backbiasing the diode, forward
current keeps flowing until recombination of the minority carriers is
' virtually completed. This again depends on the physical structure of the
diode and even upon its crystal lattice configuration. The storage time
varies widely with diode type, from a few microseconds to several milli
microseconds. Data available from the manufacturer aid in selecting the
proper diodes for the various waveshapes employed; obviously, fast pulses
cannot be transmitted by a diode which does not respond until after the
pulse has disappeared.
At this point we shall briefly summarize our mode of attack on the
nonlinear circuits analyzed in this chapter. A developmental procedure
was followed, starting with the simplest possible circuit configuration and,
step by step, increasing the complexity. In the first stage, we attempted
to gain perspective by evaluating the ,steadystate operation and com
pletely ignoring the transition from the initial to the final conditions.
Knowledge of the starting point and of the end point aids in mapping the
proper path between them. Our second consideration was the effects of
the external energystorage elements on the system response to the
drive signals. At this time simplification, even drastic simplification,
90
MODELS AND SHAPING
[Chap. 3
of the circuit avoids the necessity of concurrently treating more than a
single independent energystorage element.
Our third step concerned itself with the external capacity affecting
the switching response of the nonlinear circuit element. Again we cal
culated the response as if this were an independent problem and used
the results primarily to establish inherent circuit limitations. Finally,
we had to consider any additional complexity introduced by the circuit
element response: diode on, off, and storage time; possible resistor and
capacity nonlinearity ; and any other phenomena which might affect
the response in even minor ways.
PROBLEMS
81. In the ob gate of Fig. 3la the three signal sources have relatively high imped
ance (2 K each) and therefore, by comparison, the diodes may be assumed to be ideal
elements. The inputs are shown in Fig. 321.
(a) What is the minimum load resistor which will ensure only one conducting diode
at t = 0.3 sec?
(b) If R = 10 K, at what time will D, begin conducting? At what time will Di
cease conduction?
(c) What is the effective load (e/i) on each of the three signal sources at a time
midway between the two answers of part &?
*2
20 v
0.2 0.7 sec
15 v
1 sec,
Fig. 321
,
\h
40 v
t
Is
ec
32. The inputs of the or gate of Fig. 32 are each excited by a positive pulse: ei is
a 4^secwide 20volt pulse injected at I = and en is a 30volt 5^sec pulse applied
1 /usee later. Draw the output waveshape, labeling all important voltage values and
time constants. The circuit parameters are given below. (Make all reasonable
approximations.)
r.i = r, 2 = 500 ohms R = 20 K
d = C 2 = 10 md Co = 0.001 pi
t, = 10 ohms r, = 100 K
Fio. 322
DIODE GATES
91
88. Assume that the diodes in the gate of Fig. 322 are ideal and that they are
excited from lowimpedance sources.
(a) Sketch the output, giving all voltages and time constants if at t = a 1msec
15volt positive pulse is applied at ei and a 2msec 5volt pulse at e».
(b) Repeat part a if D» is reversed.
84. (a) Sketch the output waveshape of the circuit of Fig. 323 if the diodes may
be considered ideal.
(6) Show that the time constant of the input coupling circuit must be long compared
with the rise time of the gate and the pulse width for optimum circuit response
(nonideal diode).
t,usec „
♦ 0.01 /it
Fig. 323
86. (a) Draw the output waveshape of the and gate of Fig. 324 if both inputs
are simultaneously excited by 50/iSec pulses of 20 volts amplitude. Repeat if one
pulse is reduced to —10 volts and the other remains at —20 volts.
(6) Repeat part a for the circuit resulting when D a is removed.
+25 v
°«o
30 v
Fig. 324
86. The and gate of Fig. 36 is excited by a unit step of 30 volts at ei and a stair
case voltage which increases in steps of 10 volts every 5 fisec at e 2 . A short time after
the 40volt step is reached, both inputs simultaneously drop to zero. The diode and
circuit parameters are
r, = 200 ohms
r, = 500 K
d =0
r, = 500 ohms
C.  250 iml
R  20 K
E  25 volts
Sketch the approximate output, indicating all time constants and important voltage
values.
87. (a) If En, > ei > e s > in an and gate and if the signal source impedance is
very small, show that the gate's excitation will result in a small jump in the output
92
MODELS AND SHAPING
[Chap. 3
waveshape, after which it will rise exponentially to its final value. Calculate the
value of this jump in terms of the circuit parameters. (Hint: Treat this gate as a
compensated attenuator at t = 0.)
(6) Will such a jump appear in the ok gate? If it will, how large is it? Explain
your answer.
38. The gate of Fig. 325 is connected as shown to two inputs, a pulse of 10 volts
and 10 msec duration and a unit step of 100 volts.
(a) Find and sketch e„, labeling the waveshape with respect to important voltages
and times.
(6) If a 1,000ohm relay coil which will operate when the current reaches 1 ma is
inserted in series with D 2 , at what time will the relay be energized? When will it be
deenergized?
10 v
K
Di
9+25 v
C10K
t, msec
100 v=
10 '!
10 h
M
1 K<
Fig. 325
Fig. 326
39. Figure 326 simulates the action of a single diode in a gate slowly recovering
from a backbiased toward a conduction region. The switch is opened at t = 0, and
the various circuit parameters are r c = 500 ohms, R = 20 K, C = 0.01 /rf, Ey, =
50 volts, E a = 10 volts, and Et = 5 volts.
(a) Plot the time response of this circuit.
(6) Evaluate the slopes of the two exponential segments at the point where they
meet.
(c) Prove, in general terms, that in a circuit such as the one shown, the slopes of the
two exponential segments are always equal at the point of intersection (r c <SC R).
(d) Would the statement of part c be applicable if the diode was biased so as to
conduct at a value other than zero?
?+100v
rioK
200 K
«i° — VA — W
Ci
Fig. 327
tt°«2
:ik
310. (a) Show that the counting circuit of Fig. 327 will have an output propor
tional to the number of applied input pulses (ei) provided that they are of equal
amplitude and duration. Also discuss the possibility of erasing this stored informa
tion upon the injection of a sufficiently large negative pulse at e^.
DIODE GATES
93
(6) If the amplitude of each pulse is 100 volts and its duration is only 2 ^sec, what
size capacitor must we use so that the output will rise by approximately 1 volt/pulse
for up to 10 pulses? Sketch the output waveshape.
(c) Under the conditions of part b, how many pulses have been applied if the out
put reads slightly less than 25 volts?
311. In the controlled gate shown in Fig. 328 the controlling pulse is a 1,000jusec
20volt pulse which raises the normally negative control voltage from —20 volts to
zero.
(o) A train of 10volt 20jusee positive pulses, spaced 100 /isec apart (from the end
of one pulse to the beginning of the next one is 100 /isec) is the input to ei. Sketch
the output waveshape from slightly before the control pulse is applied until slightly
after.
(b) Repeat part a if we simultaneously apply a 25volt positive pulse train of the
same duration and period at c 2 .
(c) Does this gate offer any advantage over a similar operating version of the one
shown in Fig. 39?
Fig. 328
312. (a) Draw the circuit of a directcoupled controlled andnot gate designed
to operate on positive pulses (similar to the andoh gate of Fig. 38 except that it
should be directcoupled to all signal sources).
(b) Designate the relative amplitudes of the pulses which should be used.
(c) Sketch the input and output waveshapes and compare the results with that
obtained for the gate of Fig. 38. Does this circuit offer any advantages?
313. In an automatic milling machine, the work must be properly positioned on
the bed before the machine can be permitted to operate. The proper position is indi
cated when the states of three separate singlepole doublethrow (SPDT) switches
are changed by the pressure of the work. The center point of each switch is connected
to a constant voltage (positive), and the other terminals are available for connection
to various diode gates.
(a) Sketch a circuit which will indicate when the machine can start functioning.
(6) Sketch a circuit which can be used at the same time as the gate of part a and
which will keep the positioning mechanism in operation until the work is properly
seated.
(c) Show how you would modify the gate of part b so that the output decreases by
equal increments as each switch is activated in any sequence.
94
MODELS AND SHAPING
[Chap. 3
(<J) If you used diodes in the above parts, redraw the circuits so that they do not
require any diodes. If you did not use diodes, repeat using them as elements in the
various gates.
314. Figure 329 illustrates a simple controlled amplitude gate which will operate
for positive or negative input signals.
(a) If ei is a 20volt peaktopeak sinusoidal signal, how large must we make the
peaktopeak squarewave control signal so that the diodes will not change state as a
function of ei but only as a function of e c 1 Specify your answer in terms of the circuit
and diode parameters r„ ry, Ri, and R, (r r 58> Ri S> r/, r r 55> R, S> iy).
(6) Specify the optimum value of Ri for the maximum ratio of desired to undesired
output, when the peaktopeak control signal is 100 volts (jy = 10 ohms, r, = 100 K,
and R\ = R,).
(c) Sketch the input and output waveshapes to scale, under the conditions of
part 6, if the period of the control signal is 5 per cent of the sinewave period.
(d) Repeat part c when the input signal is zero.
«i
04
r, 4
R,<
oe.
6 e ° !
Fio. 329
316. Show the circuit of a diode array that will satisfy the following three condi
tions :
1. Select the proper 1 out of 10 channels upon the simultaneous application of pulses
in 3 out of 5 channels.
2. Reject all odd outputs upon the application of an inhibit pulse.
3. Reject all even outputs upon the application of a second inhibit pulse.
316. Construct a diode array satisfying the following truth table. All channels
marked with an asterisk are statements of an ok gate; all others are and gates. Under
which conditions are there multiple outputs?
Channel
a
6
c
d
e
1
1
1
1
2
1
1
3
1
4*
1
5
1
6*
1
7
1
1
8*
1
317. Evaluate the percentage of the control signal appearing at the output of a
diode bridge if one diode, D\, differs by 5 per cent from the other diodes. Its parame
ters are r' r — 1.05r r and r t = 0.95r/ expressed in terms of their nominal values
DIODE GATES 95
Three cases should be considered:
(a) R. = 0.2 \Avv
(b) R. =■ Vw
(c) iJ, = 5 y/r,r f .
In all cases Ri remains fixed at ^/r r r/. Can you draw any general conclusions as to
the best choice of R, in the face of the expected small variations in diode parameters?
(Let r/ = 10 ohms and r r = 100 K in your calculations.)
818. (o) Find the optimum reflected impedance through each transformer to the
common bridge circuit for the maximum ratio of desired to undesired signalvoltage
transmission (Fig. 317). What attenuation factor does the bridge introduce under
these circumstances?
(6) Further show that the conditions found in part o will also apply to a shunt con
nection. (The bridge is connected across the two transformer windings, which are
in parallel.)
319. Diodes having the characteristics of Fig. 315 are used in the bridge of Fig.
312. The maximum reverse diode voltage is 100 volts, and for safety's sake we limit
our driving signal to 75 per cent of the absolute maximum. We desire that the diode
should be biased at 1.5 volts when it is in its active region. Furthermore, the peak
input signal will never exceed 20 volts, thus ensuring that the bridge remains balanced
at all times. Find the values of R, and Ri for optimum operation under the above
conditions.
320. A diode bridge having the parameters listed below is to be controlled by a
square wave, and a total of not more than 5 per cent of the period can be devoted to
the turnon and turnoff transients. What is the fastest allowable gating signal?
r t = 100 ohms Ri = s/r r r/
r, = 250 K R. = 50 K
C d = 10 wf C„ = 250 /uif
821. In a diode bridge circuit any capacitive unbalance will cause large spikes to
appear at the output. If R, is small, calculate the time constant and relative ampli
tude of these spikes if Cn is (o) twice all other diode capacity; (b) onehalf all other
diode capacity. Take C„ as zero in this problem. (Hint: Treat in the same manner
as the compensated attenuator of Chap. 1 and use a constant input signal.)
BIBLIOGRAPHY
Brown, D. R., and N. Rochester: Rectifier Networks for Multiposition Switching,
Proc. IRE, vol. 37, no. 2, pp. 139147, 1949.
Chen, T. C. : Digital Computer Coincidence and Mixing Circuits, Proc. IRE, vol. 38,
no. 5, pp. 511514, 1950.
Hussey, L. W.: Semiconductor Diode Gates, Bell System Tech. J., vol. 32, pp. 1137
1154, September, 1953.
Millman, J., and T. H. Puckett: Accurate Linear Bidirectional Diode Gates, Proc.
IRE, vol. 43, no. 1, pp. 2737, 1955.
and H. Taub: "Pulse and Digital Circuits," McGrawHill Book Company,
Inc., New York, 1956.
Richards, R. IT.: "Digital Computer Components and Circuits," D. Van Nostrand
Company, Inc., Princeton, N.J., 1957.
CHAPTER 4
SIMPLE TRIODE, TRANSISTOR, AND PENTODE
MODELS AND CIRCUITS
An active element such as a triode, pentode, or transistor, with its
unidirectional transmission path, its internal amplification, and related
conversion of energy from the power source to the signal output, adds
a third dimension to our capabilities for wave generation and shaping.
Where previously we had only linear and nonlinear elements at our dis
posal, in this chapter we shall discuss some simple uses for a controlled
energy source which can be incorporated within, instead of remaining
external to, the waveforming network.
41. Triode Models. Consider, for example, the terminal character
istics of a typical average triode, expressed by the two experimentally
determined families of curves shown in Fig. 41. One set represents the
input grid characteristics, and the other, the plate voltampere (output
terminal) response. Any specific tube of the same type may be expected
to deviate by up to ± 30 per cent or even more from the average character
istics given by the manufacturer, as a consequence of the manufacturing
tolerances. Thus we should expect a macroscopic model to represent
reasonably well, but not necessarily exactly, the conditions expressed in
Fig. 41.
For the simplest analysis of the subsequent circuits, the model should
be a piecewiselinear one, allowing us to write a separate linear defining
equation for each region of operation. In formulating the model, we
first observe that the plate voltage has no effect on the grid characteristics
in the negative grid region and but small effect when the grid voltage
becomes slightly positive. If this secondorder phenomenon is ignored
by using the average of the curves shown to define the complete grid
response (Fig. 4la), then the approximate characteristic is identical
with that given for a diode (Chap. 2) and we can represent the grid
cathode behavior by an equivalent diode (Fig. 43). This diode's
forward resistance r c depends on the particular tube type and will gen
erally lie between 300 and 2,500 ohms; for the tube shown, it is approxi
mately 1,400 ohms. The inverse resistance is so very large that it is
almost always assumed to be infinite.
Sec. 41]
SIMPLE TKIODE AND TRANSISTOR CIRCUITS
97
Turning now to Fig. 416, the location of any individual plate character
istic, which happens to be of the same basic shape as the diode volt
ampere curve, is a function of the particular value of the applied gridto
cathode voltage. All these curves are quite similar, one to the other,
roughly parallel, and approximately equally spaced for equal gridvoltage
50 V
100
150
5 10 15
e c , volts
30
120
10
/ /
4?
>
/
'
1
/>
/ J
^>
/'
\
*
7
fi
/
4
i
'/ y
.•*>
100
400
200 300
Rate voltage e b , volts
(b)
Fig. 41. Typical triode characteristics of a 12AU7. (a) Grid voltampere terminal
response; (6) plate voltampere characteristic.
increments. We shall therefore extend the piecewiselinear concept as
previously applied to the diode and approximate the totality of plate
characteristics by a family of parallel, equally spaced straight lines
(Fig. 42). The slope of these lines should be chosen for the best match
to the original characteristics and will be roughly the reciprocal of the
plate resistance averaged over the entire plane,' l/r p . Since at any con
stant value of plate current the ratio of the change in plate voltage to
98 MODELS AND SHAPING
the change in grid voltage is
AeJ
M = —
Ac
c it*™ constant
[Chap. 4
(41)
the separation between the adjacent plate curves will be — p Ae„.
From Fig. 42 we can see that a reasonable representation of a 12AU7
requires an average n = 18 and r, = 8K With some other triodes the
H may vary from 10 to 100 and the r p from 1 to 100 K. Usually the
low values of r v are associated with the low/* tubes.
i e h "kee
50 100 150 200 250 300 350 400 450 500 550 600
[«//Ae c * Plate voltage e b , volts
Flo. 42. Piecewiselinear approximation of the plate voltampere characteristics for
type 12AU7 triode.
In the interests of clarity, the derived linear characteristics are pre
sented in a separate diagram (Fig. 42) rather than superimposed on the
actual curves. However, in order to find the piecewiselinear parameters
used to classify the particular tube type (n, r p , r c ), we would have to
refer to the manufacturer's characteristics, draw the family of straight
lines, and then calculate these parameters. After this has been done
once, there is no need to repeat the process whenever we must solve a
new problem using the same tube. Even when the piecewiselinear
characteristics are carefully approximated by eye, the deviation between
the actual and the linear curves will usually be less than 10 per cent over
the major portion of the region we are attempting to define, i.e., the first
quadrant. As expected, the largest errors occur at low plate current
where the curvature becomes rather severe.
Once the curvilinear representation is decided, we can derive a mathe
matical expression for the simplified curves drawn. The parametric
Sec. 41] simple triode and transistor circuits 99
equation describing the family of straight lines of Fig. 42 becomes
e\> ~ — fie c + r p ih n > 0, e b > (42)
where — iie e is the platevoltage intercept. Equation (42) may be
interpreted as representing the voltampere characteristics of a voltage
generator — ne c having an internal impedance of r p (Fig. 43). As the
control voltage changes by any increment, the opencircuited output
changes proportionately and the linear curve moves to a new position
parallel to itself.
An additional limitation must also be imposed to restrict further the
region defined by Eq. (42) :
e b > ke e (43)
when e e > 0. Unless the tube's operation is so restricted, the excessive
grid current flow, with the consequent lowering of the value of r c , which
occurs at low values of plate voltage and high grid voltage, may destroy
the tube; the flimsy grid structure is not designed to dissipate the heat
produced. In Eq. (43), the constant is a function of the interelectrode
spacing and the physical construction of the tube and takes on a rela
tively small range of values; k usually lies between 0.5 and 2.5. If the
tube is designed to operate under these conditions, then a new model is
necessary to represent adequately the behavior at high positive grid
voltages. The construction of such B D
a model will be deferred until needed
in Chap. 12.
It follows, from the previous dis
cussion, that the macroscopic piece
wiselinear model of the triode is the
one shown in Fig. 43. The two
diodes serve" to limit operation to ' r
the proper quadrants. The grid *
diode permits grid current flow only FlG  4_3  Triode P^ewiselinear model.
when the grid voltage is positive apd the plate diode prevents reverse
plate current flow.
When constructing the model of Fig. 43, we found it necessary to
introduce a new element, a controlled source, in order to account for
the influence of grid voltage over the plate current. The particular
signal source chosen, M e c , is a voltage generator whose amplitude is a
linear function of the gridtocathode controlling voltage. It reflects the
input signal into the output loop and thus represents the voltage ampli
fication inherent in the tube.
Under the normal operating conditions the grid diode remains back
biased while the plate diode conducts. We say that the tube "saturates ' '
«6
100 MODELS AND SHAPING [CHAP. 4
when the grid diode changes stage and conducts at e ok = e c = 0. Cutoff
corresponds to backbiasing the plate diode (4 = 0), and by writing the
voltage across this diode, the necessary condition becomes
Ec = E cMot , <  ^ (44)
where Em is the voltage appearing from plate to cathode when % = 0.
If instead we choose to replace the plate circuit by its dual represen
tation (Norton equivalent) , the voltagecontrolled voltage source becomes
a voltagecontrolled current generator. This transformation follows
directly from Eq. (42), which is rewritten so that the plate current is the
dependent rather than the independent variable. The new denning
equation is
ib = g m e e + — (45)
r v
where g m = y./r p is called the transconductance. Even though the out
a „ putcontrolled source is now a cur
M ? rent generator (g m e c ), its value is
W
still a function of the grid voltage.
S m e c{^)\ ^ g p e b Furthermore, the transformation
converted the series resistance into
a shunt conductance (Fig. 44).
Fig. 44. Alternative triode model. The twQ diodes uged in the model
reflect the physical limitations of the tube, and therefore they will remain
invariant across the change in modular representation.
Triode Amplifier. As an example of how the piecewiselinear model
is applied in the solution of specific circuits, we shall concern ourselves
with the singletube amplifier of Fig. 45a, which employs the tube whose
characteristics are given in Figs. 41 and 42. The assumption is made
that the driving signal is of a high enough frequency so that the cathode
remains completely bypassed with respect to all signal components.
Two problems require solution: first, the circuit's quiescent conditions
and, second, the inputoutput transfer characteristic.
At the quiescent point the tube is biased within its grid base, i.e.,
between cutoff and zero, and the grid is nonconducting. The gridto
cathode voltage is determined by the current flow through the cathode
resistor.
E cg = E K = h q Rt (46)
As shown in Fig. 45b, the substitution of the value of E cq , given
by Eq. (46), into the model yields a generator whose terminal voltage
is proportional to the current flow through it. But since such a response
is the same as the voltage drop appearing across a resistor due to its
Sec. 41] simple triode and transistor circuits 101
current flow, this particular voltage source can be replaced by an equiv
alent 18K resistor, /ii?3. The reduced quiescent circuit consists of the
power supply in series with various resistors; therefore
Ebi
300
69 R* + r p + ( M + 1)« 3 20 K + 8 K + (19)1 K
From the circuit we see that the other quiescent conditions are
E K = hgRz = 6.4 volts
E 2q = E bb  hcRz = 172 volts
where h 9 is as given in Eq. (47).
S 6.4 ma (47)
(48)
«iW
(a)
(b)
Fig. 45. (a) Triode amplifier; (6) piecewiselinear model.
Before beginning the evaluation of the transfer characteristic, we might
observe that the linearization of the tube characteristics results in three
linear regions of operation. Furthermore, the segments of the transfer
curve for each region are contiguous, permitting their solution by zone
superposition, i.e., the calculation of the transfer slope within each region
and the location of the boundarypoint coordinates. These individual
segments can then be superimposed on the inputoutput plane with
respect to the quiescent point to obtain the overall response.
Since only the slope is needed, it is only necessary to define an operating
region and to consider the incremental output response with respect
to some variation of the input signal falling within the defined zone. In
this way the complete output variation is contributed by the controlled
source and the' effects of the power supplies may be ignored. The three
operating regions are:
1. The large positive peaks of the input signal force the tube into grid
conduction over some function of the cycle (saturation).
102
MODELS AND SHAPING
[Chap. 4
2. The tube operates within its normal grid base.
3. The negative peaks of the input signal are large enough to drive
the tube into cutoff.
Incremental models, which hold for each of these regions, may be
derived from the piecewiselinear model of Fig. 456 by shorting all
batteries and conducting diodes and opening any branches containing
backbiased diodes. The three models appear in Fig. 46o to c. When
considering the bounds of these regions, we would have to refer back
<c>
Fio. 46. Incremental models for the circuit of Fig. 45. (a) Saturation region; (6)
active region; (c) cutoff region.
to the complete model of Fig. 45b, where the complete bypassing of R 3 ,
afforded by Cz, permits its replacement by the constant voltage Ek
[Eqs. (48)].
Region 1 : ei(t) > Ek (saturation)
In this region, since Ri J5> r c ,
*** = r» Ae i = rA Ae » = 0123 Aei
r e + R. 11.4
and from the incremental model of Fig. 46o,
nRz
Aet = —
r, + Ri
where the gridtcp'late amplification is
Ae e = 0.123 A Aei
(49)
A =
Ae2
Ae„
uRt
r T + R»
18 X 20 K
8 K + 20 K
= 12.9
Sec. 41]
SIMPLE TBIODE AND TRANSISTOR CIRCUITS
103
Note that the conducting grid introduces an additional attenuation factor
due to its loading of the signalsource impedance. The output voltage
corresponding to the boundary value of the input may be found from
Fig. 456 by setting e« = and replacing R 3 by the battery E K . The
controlled source disappears at this point, and the simple resistive net
work of Pig. 47o yields
Et, = Ek + It,r p
„ En — E K
&K + , r, ?„
Tp + «2
^ 90 volts
(410)
where h, is the saturation value of plate current.
Region 2:
E K > e,(<) >E K 
Eu
Ek
(active)
The grid now operates within its
normal base, it is nonconducting, and
it does not load the signal source.
Thus, when Ri ^> R„ Ae cl = Aei, and
Ae * = ~ , if p Ae ° = A Ae ' (4" 11 )
r P + Ri
As the negative portion of the signal
cuts off the tube, the circuit enters
the third region of operation, and by
again substituting the battery Ek for
Rz in the model of Fig. 456, we find
the cutoff value of ei(t). The resultant
simplified circuit appears in Fig. 476.
Using Eq. (44), E„ and the equivalent
input voltage become
Em,
E„ = e u (t)  E K <
eu(t) < +E K
Em — Ek
(a)
Fig. 47. limiting models for the
amplifier of Fig. 45o. (a) Tube
under saturation conditions; (6)
tube cut off.
Er
 16.3 volts
9.9 volts
(412)
The plate voltage at cutoff is simply Em
Region 3: e^t) < +E K  Ehb ~ Ek
(i.e., below cutoff)
Since the cutoff tube opens the transmission path for the input signal,
the output will remain constant at E a and, in this region, the slope of
the transfer characteristic is identically zero.
The conclusion which may be drawn from the above calculations is
that it is much easier to locate the circuit's break points than it is to find
104
MODELS AND SHAPING
[Chap. 4
the quiescent operating condition. In the triode, one coordinate of each
of these limits is already known; in one case it is where the plate current
drops to zero, and in the other it is where the gridtocathode voltage rises
to zero. We can always simplify the piecewiselinear models at these two
boundaries by eliminating the unessentials : the controlled source is
shorted at e c = 0, and the complete plate loop is opened at 4 = 0.
Finally, the remaining extremely simple network is solved for the single
unknown coordinate.
Slope=0
9.9v 6.4 v erft)
Fig. 48. Transfer characteristic of the onetube amplifier of Fig. 45.
Figure 48 illustrates the complete piecewiselinear transfer relation
ship, and as a summary of the previous discussion, all slopes and inter
cepts are labeled. The superimposed curve (dashed lines) is the transfer
characteristic as might be evaluated point by point from the manu
facturer's tube characteristics. Agreement between the two curves is
generally satisfactory, except in the vicinity of the break points, where
the pronounced curvature of the actual plate characteristics results in
wide divergence from the model representation. The slope of Fig. 48
is the value of the incremental gain at that particular value of drive
voltage.  If we are interested in linear amplification, the drive must be
restricted to keep the operating locus in the active region. Furthermore,
for the largest possible dynamic range, the Q point should be placed
halfway between grid conduction and cutoff.
The limitations as well as the advantages of the piecewiselinear
analysis become apparent once we consider how to obtain the actual
operating path of the singlestage amplifier of Fig. 45.
In the graphical construction of this locus we must first locate the Q
point, which lies along the dc load line
e h = En  i b (R 2 + R 3 )
(413)
The line defined by Eq. (413) appears superimposed on the triode char
acteristics in Fig. 49. Secondly, the actual bias point is also determined
105
Sec. 41] simple tbiode and tbansistob cibcuits
by a graphical construction. The bias equation (46) is crossplotted
on the plate characteristics, and its intersection with the dc load line
located. Finally, the locus of the timevarying output, the socalled
ac load line having the slope — l/Ri, is drawn through the Q point.
200 300
e b , volts
Fig. 49. Graphical solution of the singletube triode amplifier (the position of the ac
load line is exaggerated for the purposes of the illustration; it does not correspond to
the circuit of Fig. 45).
9+300 v
12AX7
/* = 100
rp70 K
?•<,= ! K
Fig. 410. Cathode follower for Example 41. (a) Circuit; (fc) equivalent piecewise
linear model.
For simple circuits, a graphical construction is almost as easy as the
analytical solution. In more complex circuits this is no longer true.
Even in the singletube amplifier, once the grid becomes positive we
would have to solve two sets of graphically presented characteristics:
first, those of the grid to find the actual value of e c and, next, those of the
plate circuit to find the corresponding output voltage.
Example 41. This example will consider the cathode follower of Fig. 410. We
shall solve for the cathode resistor R K that places the quiescent point exactly in the
center of the active region.
106 MODELS AND SHAPING [CHAP. 4
Solution. In the cutoff region the full supply voltage appears across the tube.
Thus the gridtocathode voltage at the boundary is given by
„ _ 500 .
H'ckc = = — a volts
M
Since the net current flow is zero, the output voltage is Ev> = —200 volts, and the
corresponding value of the input becomes
Ei„ = E cU + Eu, 205 volts
The saturation limits may be determined by setting e„ = in the model of Fig.
4106. By doing so, the controlled source is eliminated and
/, 50 °
r P +R K
The output voltage, which is now equal to the input, is given by
Ei,  E u  I b Jt K  200 volts
To obtain a second equation involving R K and h, we consider the quiescent condi
tions. When Ei — 0, the gridtocathode voltage may be expressed as
Ed,  —Ett = 200  R K It,
Substitution of this equation into the network of Fig. 4106, by way of the controlled
source, requires the multiplication of each term by it. At the Q point the controlled
source may be replaced by an equivalent resistor of Rk in series with a voltage source
of — (m + 1)E CC volts. The tube's current, found from the single loop remaining after
the substitution, is
, Eg (jx + V)E CC _ Ef.
'' r P + ( Ji + 1)R K = R K
where the approximation holds for a high/* tube [Eu, <SC —E cc (ji + 1) and R K (fi + 1) »
»■,]. In order for the quiescent point to lie in the center of the active region h, = h./2.
Substituting values and solving results in Rk = 280 K. The previously unknown
values can now be evaluated.
lb, = 0.715 ma
Ei, = +2.5 volts (from the more exact solution for h,)
Ei. = Eu  200 volts
The piecewiselinear model has not only freed us, in our calculations
of the tube quiescent point of operation, from any dependency on graph
ical construction, but also has contained within the model a diagrammatic
representation of the complete range of tube behavior. All incremental
models were directly derived from the single piecewiselinear model,
and the particular constraints imposed on the regional models also
explicitly appear. If, in any region of operation, a more exact calcula
tion as to the incremental behavior is needed, we can always return to
Sec. 42] simple triode and transistor circuits 107
the actual tube characteristics and calculate the incremental values of
r p and m for the microscopic portion of the voltampere plane under
review. This is particularly necessary when operating at low plate
currents where the low slopes of the actual characteristics indicate a
large increase in r r . In this same region, adjacent curves begin crowding
together, with a consequent decrease in the incremental ju (Fig. 416).
The piecewiselinear model cannot account for these variations in
tube parameters since the basis of its construction was the linearization
of the tube characteristics. Over most of the plane, the values of /i and
r p are not much different from the constant values assumed in the linear
curves of Fig. 42. We may conclude that the macroscopic model is
generally valid for largesignal applications, that it is reasonably valid
for smallsignal analysis over much of the operating region, and that it
represents a convenient firstorder approximation over the remainder
of the plane.
42. Triode Clipping and Clamping Circuits. From a study of the
transfer characteristic of Fig. 48 we conclude that, besides functioning
as a linear amplifier within its active region, the nonlinearity of the
triode may also be utilized as a doubleended clipper. Each break
point corresponds to a change in state of one of the diodes used in the
piecewiselinear representation (Fig. 43); the clipping of the positive
input peak occurs as the grid is driven into conduction, and the negative
peak is clipped in the plate circuit as the controlled source cuts off the
tube. We generally reflect the cutoff action back to the grid and speak
of the spacing between the two break points, in terms of the gridto
cathode voltage, as the grid base of the particular tube. Control over
the width of the transmission region is rather awkwardly effected; either
the cutoff point must be changed by adjusting the plate supply voltage
or we must replace the tube with one which will operate within a smaller
or a larger grid base (i.e., a different value of m).
In spite of this limitation, the triode is widely used as a clipper because
advantage may be taken of its internal amplification to develop a large
clipped output from a relatively small driving signal. For example,
if a sinusoid having a peaktopeak amplitude much larger than the grid
base is chosen as the input, then the clipped output will be a fair approxi
mation of a square wave. Even better squaring results when a second
stage is cascaded for further clipping and amplification. Simple differ
entiation converts the square wave into a pulse train which might be
applied to other circuits for synchronization or control.
The undesirable information transmission occurring in the saturation
region may be minimized by inserting a very large resistor in series with
the control grid. But since the input capacity of a triode is quite large,
50 to 300 ppi, the resultant long input time constant adversely affects
108 MODELS AND SHAPING [CHAP. 4
the rise time of the stage. For this reason, clipping is usually limited
to the cutoff region and a compound circuit, such as the cathodecoupled
amplifier shown in Fig. 411, would be used instead for doubleended
operation. Here each clipping region corresponds to one of the tubes
being driven into cutoff, and therefore no special provision need be
made for operation in the saturation region.
Qualitatively, this circuit operates in the following manner: tube Ti
is a cathodefollower input stage, injecting the input signal into the
cathode circuit of T 2 . The negative
bias E ee serves to set the proper
quiescent conditions. At very large
positive values of ei, the positive
commoncathode voltage cuts off tube
T 2 , and through doing so destroys
the transmission path from the input
to the output. In this region the
slope of the transfer characteristic
becomes zero. As the input signal
falls, Ti enters its active region and
Fig. 411. Doubletriode clipping cir the circuit functions as a cathode
cuit— cathodecoupled amplifier. coupled amplifier. Linear ampli
fication ceases when the falling signal
voltage finally cuts off T\, again interrupting the transmission path and
again making the transfer slope zero.
When the input is large enough to cut jT 2 off, the model of Fig. 412o
holds and will be used to calculate one of the boundary values. We must
first write the following gridcircuit defining equation:
e c i = ei — itiRi — E ce (414)
Substitution of Eq. (414) into the model of Fig. 412a results in the
replacement of the controlled source ne c i by two voltage sources —nE cc
and nei and an equivalent resistor fiR 3 . Thus the effect of the driving
signal is brought to the fore and the plate current becomes
. = E» — (ft + 1)E CC + m^i
r, + (m + l)B t
But in order to ensure that T 2 is off and that our argument as to the
circuit operation is valid, the gridtocathode voltage of T 2 must be
equal to or below cutoff. By taking the external loop voltages, from the
grid to the cathode, we obtain
e« = E cc E t < Ebb " Ec ° ~ Es (415)
Sec. 42] simple triode and teansistoe ciecuits 109
where E% = %\Rz and the platetocathode voltage of T 2 is Eu> — E cc — E 3 
Solving Eq. (415) for E t and equating to i b iR 3 yields
E%
E»  (ji + l)E cc _ En + nEn  0* + l)E a
M+ 1
r p + ( M + l)fi,
Rs
The upper bounding value of the drive signal En is readily found by
solving the above equation.
The location of the other break point would be found in a similar
manner by directing our attention to the model of Fig. 4126. Again
the additional information needed for a complete solution is furnished
by the cutoff equation of the nonconducting tube T x . The two equations
involving E% are
Ez
e c i
ibiR3 —
E»  0» + l)E c ,
R,
Ei
r P + fl 2 + (/i + 1)R,
Em, — E cc — E%
Ecc — Ei <
The simultaneous solution of these equations for En, the minimum value
of the input which can be tolerated before Ti is driven into cutoff, com
pletes the solution of this problem.
Triodegr idcircuit Clamping. A
further circuit application of the
triode, with its twodiode repre
sentation (Fig. 43), is as a clamp.
Both the grid and plate circuits
will individually clamp an external
signal in exactly the same manner
as the diode clamps of Chap. 2, but
because the equivalent diodes used
in the triode model are irreversi
ble, only negative clamping is
permitted.
By properly adjusting the cath
ode bias resistor, the gridcircuit
clamp level may be varied from zero to a positive value equal in magnitude
to the tube's cutoff voltage. Since the triode grid has an almost infinite
reverse resistance, this circuit would require a resistor, shunted from
grid to cathode, in order to ensure the proper charging of the clamp
capacitor (Sec. 25).
This same clamping action is widely employed as a means of establish
ing the grid bias of the tube. In Fig. 413, the grid conduction on the
positive peaks of the applied signal results in a current which charges C
to the positive peak of the input voltage. Except for the short interval
(a)
Fig. 412. Piecewise models of the circuit
of Fig. 411. (a) ?\ on and T 2 cut off;
(ft) Ti on and Ti cut off. (Note: E cc < 0.)
110
MODELS AND SHAPING
[Chap. 4
when the dissipated energy is restored in the capacitor, the circuit auto
matically adjusts itself so as to avoid operating in the positive grid
region. The negative grid clamping is reflected, through the controlled
source, into effective positive platecircuit clamping as illustrated by the
waveshapes of Fig. 413. Note that operation over the complete grid
base is guaranteed before the tube cuts off. Of course, a small amount of
distortion appears as a consequence of the small grid conduction angle,
but if this can be tolerated, then this circuit represents an extremely
simple way to selfbias a tube and still ensure the maximum possible
voltage swing.
M
rw
vw
Fig. 413. Gridcircuit clamping and resultant waveshapes.
We might further observe that any capacitorcoupled amplifier will
exhibit a measure of gridclamping action when the input signal over
drives the grid, i.e., when the positive peak of the input is greater than the
cathode bias voltage. The additional bias component produced shifts
the operating point away from the positive grid region, and provided that
the tube is not driven into cutoff, the actual distortion will always be less
than expected on the basis of a simplified analysis.
43. Triode Gates. Almost any configuration of triodes, coupled
together by a common load resistor in either their cathode or their
plate circuit, will serve as an and or an or gate (Figs. 414 and 416).
Since the individual tubes are essentially in parallel, the basic considera
tion for proper functioning as an or gate is that each input pulse should
be able to make its presence known at the output. If the tubes are
normally biased below cutoff, then the conduction of any one tube,
brought about by an input pulse, will operate the gate and produce a
change in the circuit state.
On the other hand, in an and gate the effects of a single excitation
pulse must be swamped out by the normal conditions of the other
parallel tubes. In biasing them in their saturation regions the aggregate
effect of the load current contributed by all the tubes would not be
seriously disturbed if, upon the injection of a negative pulse, a single tube
Sec. 43] simple teiode and teansistoe cieguits 111
switches from saturation to cutoff. The major change of circuit state
will occur only when all the tubes are driven off.
Triode OR Gate. We now propose to examine the operation of the
commoncathode or gate of Fig. 414o. Besides calculating the steady
state response from the piecewiselinear model, we are also interested
in finding the rise and decay time constants of the output waveshapes.
All tubes are normally maintained completely cut off by returning
the grid resistors to a large negative bias (E cc <K 0). The input pulse
is applied through the R\C\ time constant, which may be assumed long
Fig. 414. (a) Commoncathode triode ob gate; (6) piecewiselinear model of a single
on tube.
enough, with respect to the duration of the driving signal, so as not to
introduce any additional shaping. We shall further assume, purely for
the ease of calculation, that the single pulse applied to any grid will drive
that tube to the verge of saturation.
The gridtoground voltage corresponding to saturation may be found
by setting e e = in the piecewiselinear model of Fig. 414b. Solving
the reduced network for the equivalent input and output voltages leads to
E a = Eht —
Rl
r p + Rl
Ehk
(416)
However, the grid was initially at E cc , and since we wish to raise it to
the value given in Eq. (416), the input pulse height would have to be
B*
Ei,, — Ea
112
MODELS AND SHAPING
[Chap. 4
The output due to smaller or larger inputs could be calculated, quite
easily, once values are assigned to the circuit parameters.
The output rises toward this steadystate value with a time constant
determined by the output load and the internal impedance of the' tube.
Since, in the circuit configuration shown, the controlled source includes
the output voltage as one of its components, the value of the output
impedance is not immediately obvious upon inspection of the model.
By proceeding from the basic definition, i.e., the output impedance is
equal to the ratio of opencircuit terminal voltage and the shortcircuit
current, its evaluation is straightforward. Furthermore, the dc sources
may be ignored since they only set the quiescent operating point.
By removing the external load Rl  C„ the opencircuit output voltage
of Fig. 414b is given by
e
tie c = m(«i — e )
(417)
In finding the shortcircuit current, we return the cathode directly to
ground, thus removing the dependency of the controlled source upon any
voltage appearing in the output loop.
Therefore
(418)
Solving Eq. (417) for e„ and dividing
the result by Eq. (418) yields
z ;
1
M+ 1
g m
(419)
Fig. 415. Input and output pulse
appearing in the commoncathode or
gate of Fig. 414a.
ance, on the order of 400 to
constant is quite small:
Tl
700
The above approximation holds for a
high/* tube (i.e., where m + 1 = /*)•
The single tube considered to be in
its active region is operating as a
cathode follower, and Eq. (419) ex
presses its very small output imped
ohms. Consequently, the rise time
( Bll iTTi)
c,
where C, represents the total stray capacity appearing across the output.
Upon removal of the excitation, the tube cuts off and the output
decays back toward zero with the new, much longer time constant
Tj = RlC,
Sec. 43]
SIMPLE TRIODE AND TBANSISTOE CIRCUITS
113
which is simply that of the external load. The input and output wave
shapes sketched in Fig. 415 illustrate the pulse distortion introduced
by this gate. Usually Rl is kept small so that the trailing edge will
not be smeared excessively and therefore falsely indicate the presence
of a gating signal after its removal.
The particular gate of Fig. 414 was designed for positive pulse excita
tion; for gating negative pulses we would turn to a common plate con
nection and apply the negative input pulses in the cathodes of the
individual tubes. Under these circumstances, the triode would amplify
the input pulse, but with its larger output impedance it would also
introduce a greater degree of distortion in the initial rise of the output
pulse.
iE cc (E cc >0)
Fig. 416. Common plate connection — and gate.
Triode and Gate. An alternative mode of gate operation is typified
by the and gate of Fig. 416. The grids, returned to the positive bias
Ea, through R lt are normally maintained in their saturation region, with,
however, their selfclipping action limiting the maximum positive grid
excursion to
E„ =
E a
r e
Ecc^O
(420)
Ri + »\s " Ri
where Ri » r .
Upon the excitation of any one grid, by a negative pulse, that par
ticular tube will cease conduction, and since its plate current no longer
flows through R L , the composite plate voltage rises slightly. Provided,
however, that R L » r„ this change in voltage will be almost negligible.
Only after every tube is simultaneously cut off does the plate voltage
change by an appreciable amount. It rises from its initially low satura
tion value of
Eu^
Rl + r„/n
Em
(421)
which is found from the piecewiselinear model holding when n tubes are
saturated — to the final value of E».
114
MODELS AND SHAPING
[Chap. 4
One obvious peculiarity of this and gate is that it produces a positive
output upon excitation by negative input pulses, and if we wish to drive
a second, similar gate, a buffer amplifier must be inserted to invert the
pulse. If the circuit is modified so that positive driving pulses are
injected into each cathode, the gate output will be an amplified pulse
of the same polarity as the input.
Amplitude Gates. Triode amplitude gating is predicated on the con
trol signal (normally injected at the cathode) shifting the operating
point of the cutoff tube into its active region. At this time the amplifica
tion from grid to plate permits the transmittal of the information signal.
The basic gating behavior is illustrated by the simplified circuit of Fig.
417a. But to avoid loading the controlsignal generator by the low out
put impedance seen in the cathode circuit, the control signal might well be
°E K
(a) (6)
Fig. 417. (a) Basic amplitude gate; (6) cathodecoupled controlled gate.
coupled through a cathode follower. The resultant circuit (Fig. 4176)
is of the identical configuration with the cathodecoupled clipper of
Fig. 411, with, however, signals applied to both grids.
Under normal conditions, the positive dc level of the control signal
raises the commoncathode voltage to a value that is high enough to
ensure biasing Tt well below cutoff. For optimum performance, the
response of this gate should be independent of the controlsignal ampli
tude; to this end, the gating signal simply drives Ti into its cutoff region,
allowing T% to function as a normal amplifier. If the above conditions
are satisfied, the calculation of the output voltage, with respect to the
gated signal e„ may be carried out without reference to TY
The control pulse still makes its presence felt at the output, although
somewhat indirectly. As seen in the waveshapes of Fig. 418, the
abrupt change of state of T2, which takes place when the tube enters
into its active region, effectively superimposes the gated output on a
t
Sec. 43] simple triode and transistor circuits 115
pedestal having the same duration as the gating pulse. Under many
circumstances such an additional term would be intolerable. It follows
that in order to eliminate the output pedestal, we must maintain the
current flow through the output load invariant with respect to the state
of T 2 . We accomplish this by
adding a third triode (T 3 of Fig.
419) which is gated off at the same
time Ti is gated on. Since T%
normally conducts and since it
timeshares the output load, we
can adjust its plate current, by
varying JB«, so that the output
voltage remains constant across
the change of circuit state. The
application of a negative gating
pulse drives both T\ and T x into
cutoff. These tubes are effectively
(a)
Ekb
t
eo
M
m/wT
AAA/WWWWWWVWl t 
(b)
Fig. 418. Cathodecoupled gate wave
shapes, (o) Control signal; (6) input
signal and gate output.
removed from the circuit, and the transmission path, now established
through Ti, is unaffected by any other triode. As the waveshapes testify,
the output voltage no longer changes abruptly; the pedestal is now
missing.
[ftp
#
£
t
A/V
: 3 4 \AA/Vr
Fig. 419. Circuit for elimination of the output pedestal.
In concluding this summary of triode gates, we might note that the
controlled source, which is inherent within the triode, affords a much
greater degree of isolation between the input and the output than was
possible with diode gates. Moreover, this isolation greatly reduces or
eliminates completely any interaction between the various driving signal
sources.
116 MODELS AND SHAPING [CHAP. 4
44. Transistor Models. An npn transistor consists of two junc
tions of n material, either silicon or germanium, to which ntype impuri
ties have been added. These are formed on each side of the thin layer
of p material constituting the base of the transistor. One junction is
normally forwardbiased with respect to the base structure, in this case
by making its polarity negative (v<* < 0), and it serves as the emitter.
We might observe that structurally the emitterbase junction forms a
forwardconducting diode. The remaining junction becomes the col
lector, and for proper transistor operation, it should be backbiased (posi
tive with respect to the base). Furthermore, the collector junction
current is approximately equal in magnitude to the emitter current (Fig.
420c), but its voltage drop will be much larger. Since most of the
transistor power is dissipated at the collector, its junction must be com
mensurate in size. However, special transistors have been designed for
bidirectional transmission, and in these the particular junction which
would function as the emitter would be determined solely by the voltages
present in the circuit. At any instant whichever junction happens to
be forwardbiased becomes the emitter, and if the other is backbiased,
it becomes the collector.
A transistor's characteristics are determined by the particular material
used, i.e., germanium or silicon, and by the nature of the junction formed.
The major differences in the voltampere curves appear at low collector
voltages. These variations are relatively minor, and therefore the models
which shall be derived below are adequate representations of all types of
transistors.
Figure 420a, the schematic representation of the npn transistor, is
presented in order to define the circuit polarities which we shall employ
in the remainder of the text. The ready availability of complementary
units, i.e., both npn and pnp transistors, creates some confusion when
a current direction is arbitrarily assigned. In an effort to avoid any
ambiguity, we shall choose the emitter arrow as the reference and assume
all current flows in the direction it points. If the emitter current flows
out (as shown in Fig. 420a for the n~pn unit), then the base and collector
currents flow into the device. Alternatively, in the pnp transistor the
emitter arrow is pointed in the opposite direction, into the transistor, and
all voltages and currents would be reversed. Thus the transistor may
be considered as a simple node where
i. = h + i. (422)
Before attempting construction of a piecewiselinear model for the
transistor, we must satisfy ourselves as to the suitability of such a repre
sentation. Following much the same procedure used in deriving the
triode model, our starting point will be the examination of the empirically
Sec. 44]
SIMPLE TRIODE AND TRANBISTOB CIECUITS
117
determined transistor characteristics, which are either furnished by the
manufacturer or obtained by measurement in the laboratory. Even
though gross approximations may be acceptable in replacing the actual
characteristics by a family of straight lines, the linearized curves should
= U €*
°cb ~~.
(a)
10 v lv
0.1 0.2 0.3
Emitter voltage va
(b)
8
, 1
I e —8 ma
1
• 6
6
■*■
c
fc
" 4
4
i
8
2
2
If
flco
2
20
24
8 12 16
Collector voltage va, volts
(c)
Fio. 4r20. Typical commonbase characteristics — junction nrjm transistor, (a)
Schematic showing current flow and polarity; (6) input voltampere characteristic;
(c) collector characteristics.
still lead to results consistent with the transistor's physical behavior;
that is to say, the actual curves must be readily representable by straight
line segments and the equal increments of the controlling variable should
produce roughly equal changes in the controlled term.
The most superficial glance at the voltampere characteristics of the
typical rirp^n junction transistor (Fig. 420) convinces us that the device
118
MODELS AND SHAPING
[Chap. 4
itself behaves in an almost piecewiselinear manner. Therefore we
expect that the model representation will bear a closer correlation to the
actual characteristics than it was possible to achieve with the triode.
On the other hand, since the difficulties encountered in the manufacture
of transistors manifest themselves in a wider divergence of characteris
tics, their actual response in a circuit would probably differ, by a com
parable amount, from the ideal. By again directing our attention to
Fig. 420, we would draw the following additional conclusions. First,
the transistor exhibits currentcontrolled behavior in contrast to the
voltagecontrolled response of the triode. Secondly, we see that the
collector curves are almost constantcurrent lines, with the particular
operating path determined by the emitter current. Finally, from a
10 
.
10
* — 8
6
1
tfAi',,
... 2
t
i i i 1
1 hP
5 10 15 20
V c b, volts
(b)
Fig. 421. (a) Linearized transistor input characteristic; (6) piecewiselinear collector
characteristics.
V„0.1 0.2
Veb, VOltS
(a)
macroscopic viewpoint, the effect of the collector current on the input
response is not very pronounced; for the initial discussion the feedback
from the output to the input may be ignored.
Commonbase Model. In finally replacing the actual characteristics
by the linear segments needed for the largesignal model, we shall approxi
mate the set of input voltampere curves (Fig. 4206) by the single line of
Fig. 42 la. Its equation is
»e6 = (V + »>n) U >
(423)
Equation (423) corresponds to a biased diode having a forward resistance
of rn. Often this term is referred to as the largesignal input resistance
(groundedbase connection).
The collector characteristics are replaced by a family of parallel
straight lines having a spacing determined by the emitter current. Their
Sec. 44]
SIMPLE TRIODE AND TRANSISTOR CIRCUITS
119
parametric equation, which we write in terms of the variable of interest,
becomes
i, = f(v A ,i e ) = gja + ai, i e > and iu > (424)
The above defining equation (424), representing the response of the
output circuit, is analogous to that written for the triode plate circuit
[Eq. (42)]. The roles of current and voltage are simply interchanged,
and we shall see that this leads to the dualmodel representation. With
respect to the original characteristics of Fig. 420c, the linear slope is
g c (the opencircuit collector admittance) and the current intercept is
ai,. Moreover, the forwardcurrent amplification factor
At, J
At« Iamo
(425)
fulfills a similar role to that performed by n in the vacuum tube. Both
coefficients may be found by measuring the collector current, first,
with an applied collector voltage and the emitter opencircuited (g c ),
and second, with an applied emitter current and the collector short'
circuited (a).
"eb
o ci
Fig. 422. Fiecewiselinear model — npn junction transistor.
We can construct the piecewiselinear model shown in Fig. 422 from
the above defining equations. The forwardcurrent transmission appears
in the output circuit as the current generator ai,. Diodes are inserted
to restrict operation to the appropriate regions as well as to represent
the transistor's behavior under saturation and cutoff conditions. We
consider that the transistor is cut off when the input diode is backbiased
and that it is saturated once, the collector diode conducts. These con
ditions lead to the following limits for the active region:
v* < — Ve Vcb>0
(426)
When the external emitter supply voltage is relatively large, the
error in subsequent calculations due to omitting the small constant
emitter drop will be insignificant. Moreover, the treatment of the input
circuit as a constant resistance will greatly simplify any analysis. Except
120 MODELS AND SHAPING [CHAP. 4
when using low supply voltages, the small drop will be ignored and the
emitter will be assumed to conduct at zero volts. The values of the
parameters of the junction transistor normally he within the following
bounds:
r u = Ha ^ 10 to 100 ohms
Vo = 0.05 to 0.4 volt
ffc = Ha, =  ^ 10* to 10 7 mho
a = h fh £* 0.95 to 0.99
The h parameters tabulated above are those normally given by the
manufacturer; the first subscript indicates the terminal under considera
tion (i for the input and o for the output), and the second subscript, the
common grounded terminal (b for the base and e for the emitter).
If we refer back to the collector characteristics, we shall observe
some small collectorbase current flow even when i t = 0. This term
is ho, a temperaturedependent current which is due to the minority
charge carriers present in the transistor material and which might be
equated to the reverse current flow of a semiconductor diode. At room
temperatures I e o would be only a few microamperes, but since it roughly
doubles for every 10°C rise in the ambient temperature, it would become
important at elevated temperatures. Its equivalentcircuit representa
tion would be an additional current generator in parallel with the con
trolled source shown. In the interests of simplicity, I c o will be omitted
from the following discussion; the reader can always include this term
where necessary.
Commonemitter Model. Many transistor circuits exist where the input
current is injected into the base and where the emitter replaces the base
as the reference terminal. Consequently, it would be of interest to
construct an alternative largesignal model in which the base current is
the controlling term.
This transformation follows directly from the model of Fig. 422. We
might first note that
4 = i.  ic = (1  <*)i. (427)
Solving Eq. (427) for *, and multiplying the result by a yields
cd. = r ^— i b = fa (428)
1 — a
where /S is identified as the basetocollector currentamplification factor.
Since a is close to unity, taking on the values given above, normally
ranges between 20 and 100.
To complete the transformation from the emittercontrolled circuit
of Fig. 422 to the basecontrolled model of Fig. 423, we must ensure
Sec. 44] simple tkiode and tranbistok circuits 121
that all the remaining terminal characteristics of the two models are
identical. In both, the emittertobase voltage drop must be the same
for a given emitter current. For the groundedbase circuit,
v * = — (Vo + iSu)
In the groundedemitter configuration
vt. = »* = +Fo + wit (429)
Substituting the value of t» given in Eq. (427) into Eq. (429),
vt. = Vo + (1 — «Kit.
Thus, by comparing appropriate terms,
_/ _ *"u
1  a
 (l + Mm
(430)
Under saturation conditions (conduction of the collector diode) the
controlled source is shorted, is removed from the circuit, and r' n reduces
to rji.
ri'ifl+UMi
w\
rOi
"V
V
VW
w
»a
Fig. 423. Largesignal transistor model — commonemitter equivalent circuit.
The single remaining unknown term in the new model is the relative
value of the output conductance tt. Once we set i, = 0, in the circuit
of Fig. 422, we can then write the simple equation
»<* = tefc
h.
(431)
where the small constant term To is neglected. From Fig. 423, under
the same conditions of an opencircuited emitter,
lb = — tc
But the current source in the output loop is controlled by the base cur
rent flow, which in this case is —i c . Substitution into the model of
122
MODELS AND SHAPING
[Chap. 4
Fig. 423 results in the following output equation:
Vce = t«(l + /3)rd
(432)
Since the emittertobase voltage is extremely small, «<* = t>„. By equat
ing Eq. (431) to (432), we can find the internal impedance of the base
controlled current source. It becomes
r.
rj = (1 — a)r c =
+ 1
(433)
and rd is only onetenth to one onehundredth part of r c .
The reverse collector current / c o also flows through the base circuit
and with no input
ib — ~ Ico
This term will also be multiplied by the base amplification factor 0.
For consistency of current flow, the controlled source must be shunted
by an equivalent current generator of (1 + p)I c o The parallel combina
tion of these two generators adds up to the actual current flow Z c o.
10
£g
OfiA
V76
\
50_
\'t
6_
_10C
_
75.
j0_
1
25
■
1
3
16
20
24
12
V C e, VOltS
Fig. 424. Transistorcollector voltampere characteristics — commonemitter con
figuration.
The nonlinear attributes of the model must remain invariant under
the circuit transformation, and therefore the two diodes remain in the
emitter and in the collector arms. This is consistent with physical
behavior since the emitterbase and the collectorbase circuit each super
ficially constitute a diode, one forwardbiased and the other backbiased.
If we started the construction of the transistor models from the volt
ampere characteristics which are measured for the commonemitter
configuration (Fig. 424), the model resulting would be identical with
Sec. 44]
SIMPLE TRIODE AND TRANSISTOR CIRCUITS
123
Fig. 423. We note that the slope of the collector curve is much steeper
than that of the commonbase circuit, thus verifying the result of Eq.
(433). Moreover, as we anticipated in Eq. (428), the change in collector
current with respect to changes in base current is much larger than with
respect to the former controlling changes in the emitter current.
The comparative h parameters for the terms in the model of Fig. 423,
together with their range of values, are
r' u = H ie = 100 to 1,000 ohms
1
9d =
Td
Ho, = 2 to 100 jtmhos
= H„ = 20 to 100
Incremental Models. When dealing with small signals it is occasionally
necessary to account for the effects on the input circuit of the changing
collector voltage. In the largesignal
model this factor was neglected, lead
ing to the simple input circuit of Fig.
422. To take it into account in the
incremental model, we can insert a
voltagecontrolled source as shown in
Fig. 425. The reversevoltage ampli
fication h^, is very small, on the order
of 10 6 to 10~ 8 . For a more accurate
representation, the other parameters (a, r e , and rn) may be reevaluated
at the operating point.
For the model of Fig. 425, the two defining equations are
t><* = — riii, + have (434o)
ic = od e + g<Pd> (434b)
where the voltage and current terms represent incremental variations
about the operating point. From Eq. (434a) we see that the small
reversevoltage transmission is defined as
Fig. 425. Hybrid transistor model —
commonbase connection.
hrt, =
AVeb
(435)
Inspection of Fig. 4206 would indicate that h T b increases with decreasing
collector voltage and decreases with decreasing emitter current. In any
calculations, it must be evaluated at the known operating point.
If the twosource hybrid model of Fig. 425 is replaced by the equivalent
T network of Fig. 426, one of the controlled sources will be eliminated.
Yet its effect is still present, represented instead by the commonbase
124 MODELS AND SHAPING [CHAP. 4
resistor »v The equivalents of the two models are found by solving the
terminal response under identical operating conditions.
For a shortcircuited output, the input equation of the hybrid model
[Eq. (434a)] reduces to
f* = — r n i.
The equivalent equation found from Fig. 426 is
»* = [r.+ (1 «)*]*. (436)
where r' c is very large compared with r b and may be neglected. Thus
r, + (1 — a)r h = r u
and for a very close to unity, r, very closely approximates ru.
Fig. 426. Incremental T model — com Fio. 427. Incremental T model — com
monbase connection. monemitter connection.
By solving for Vj, from the models of Figs. 425 and 426 at i. = 0,
h <» = Hr? = p (437)
r h + r c r e
Following the same procedure with respect to the collectortobase
terminal response and remembering that r' c » r b , the remaining equiv
alents are
r' c S r c and a' = a (438)
The values of the two new parameters generally lie within the range
r b , from 100 to 400 ohms
r„ from 10 to 50 ohms
and in much the same manner as htb, n varies widely with emitter current
and collector voltage. It may decrease by a factor of 2 to 10 as we
drive the transistor from close to cutoff to close to saturation.
The commonemitter T model (Fig. 427) will be almost identical
with the commoncollector model, with the major difference appearing
in the collector branch. As in the equivalent hybrid model, r<j replaces
Sec. 45] simple triode and transistor circuits 125
r c and /3 replaces a. Furthermore
r' n £* r h + (1 + /S)r.
Largesignal T Models. As an alternative to developing the incre
mental model from the largesignal representation, we can begin by
examining a small section of the voltampere plane and, as we expand
the area under review, modify the incremental circuit toward a large
signal model. Our starting point might be one of the T circuits of
Fig. 426 or 427, and two limiting diodes would be added to restrict
the permissible operating region. If necessary, a bias battery Vo can
also be inserted in the emitter circuit. Since a model is, at best, a reason
able approximation of the actual response, if average values are chosen
for the T parameters, the circuits of Fig. 428 will be as serviceable as the
hybrid models of Figs. 422 and 423.
o — 4_AAA
(") (b)
Fig. 428. Piecewiselinoar T models, (a) Commonbase connection; (6) common
emitter connection (Fo and I c o terms omitted).
45. Simple Transistor Circuits. The transistor, even more so than
the triode, is readily adaptable for clamping and clipping applications.
Since structurally it may be likened to a pair of diodes back to back, we
might use either one independently in any of the circuits of Chap. 2.
As a single diode is more economical, this is at best a dubious choice.
But if advantage is taken of the current source shunting the collector
diode to establish the location of the clamping level or the clipping point,
as in Fig. 219, then the necessity of introducing an external bias gen
erator will be avoided. Furthermore, since complementary transistors
exist, npn and pnp, we are no longer restricted, as we were with the
triode, to unidirectional operation.
It is of interest to calculate the overall transfer characteristics and,
in the process, to delineate the various regions of transistor behavior.
At the same time an attempt will be made to simplify the twoloop models
of Figs. 422, 423, and 428 to the point where they become almost
ridiculously simple. Of course, the approximations made must be kept
126 MODELS AND SHAPING [CHAP. 4
in mind because many transistor circuits may not lend themselves to this
treatment. The many others that do, including most of the circuits
which will be treated in later chapters, justify our spending some little
time on this topic.
Fig. 429. (a) Transistor amplifier circuit; (6) piecewiselinear model.
Consider, for example, the transistor amplifier circuit of Fig. 429o.
In drawing the T piecewiselinear model (Fig. 4296), we also converted
the circuit, external to the base, to its Norton equivalent, where
Ri = R,  Ra
and
I _ Eg
Ra
Before beginning the analysis, we might note that the percentage of the
driving current which actually flows into the base depends on the relative
magnitudes of the transistor base input impedance Z\, and the external
base load JRi.
R,
Ri + Zb
ii{t)
(439)
It thus seems logical to evaluate the overall transmission in two steps:
first, the direct transmission from the base terminal to the output and,
second, the transmission factor relating the driving source and the base
current. The overall forward transmission is the product of these two
terms. Moreover, since the bias current h only shifts the transfer char
acteristics with respect to the zero point, it might be just as easy initially
to ignore h and, after the solution is complete, then consider the shift in
axis introduced by this constantcurrent term.
Sec. 45]
SIMPLE TRIODE AND TRANSISTOR CIRCUITS
127
The first operating region with which we shall be concerned is the cut
off zone, characterized by the backbiasing of the emitter diode. Figure
430o presents the reduced model holding in this region. The input
circuit defining equation will enable us to locate the boundary.
v b . = ibH + OS + 1)4^ + Riib + #i*
= ibln + OS + l)r„ + R2] + Etb
(440)
But the reader may verify from Eq. (433) that (/3 + l)r d = r c . It now
seems reasonable to neglect the small voltage drop appearing across r b
Fig. 430. (a) Model defining cutoff region; (6) approximate representation.
while considering the bounds of this region (r c ^>r b ). For the emitter
to be backbiased v be < 0; solving Eq. (440) establishes the cutoff region,
in terms of 4, as
' Evb ~" (441)
4<
r.+ ffi
=
The current flowing in the base under cutoff conditions is simply the
reverse current through the backbiased collectortobase diode.
The input impedance of the transistor is the coefficient of i b in Eq.
(440). Since the predominant term is r c , which is usually very much
larger than any external load appearing at either the collector or the base,
practically all the signal source current will be shunted through Ri.
Transmission from the input to the output is completely negligible, and
for simplicity, this path may be replaced by an open circuit. The model
of Fig. 4306 results, and we have seen that it represents, quite ade
quately, the transistor's behavior under cutoff conditions.
Next, in considering the active region of transistor operation (Fig.
43 la), we note that for maximum load current, the external load should
be very small compared with the internal impedance, that is, R 2 « r d .
Under this condition the current through R 2 will be $i b . Along with
neglecting the current flow through r A , the resistance itself will be removed
128
MODELS AND SHAPING
[Chap. 4
from the equivalent circuit. A current of t 6 (l + 0) flows through r„
and the input equation for this region becomes
vu = ibn + (1 + /3)r«4
= 4[r„ + (1 + /3)rJ (442)
The interpretation given to Eq. (442) is that any resistance in series
6t
►^l V
il(t)
(b)'
Fio. 431. (o) Model defining active region; (b) approximate representation.
Fig. 432. (a) Model defining saturation region; (6) approximate representation.
with the emitter is multiplied by (1 + 0) as it is reflected into the base
input circuit. From Eq. (439),
Ri
and
% r b + (1 + $)Y. ;+ i?i
ic = /34 =
»'i(0
H + (1 + 8)r. + «!
»'i(0
(443)
For the maximum possible current gain, i?i 55> rt + (1 + 0)»V Upon
satisfaction of this necessary inequality, both the base and emitter
resistors are replaced by short circuits and the trivial model of Fig. 4316
results.
The upper bound of the active region occurs when the collector voltage
drops to zero. From the simplified model of Fig. 4316,
v„ = En  pibR 2 >
Sec. 45] simple teiode and transistor cibcuits
Thus the limits of the active region are
En,
< % <
fiRi
129
(444)
As t'x(t) continues to increase, the circuit enters upon its saturation
region. The collector diode con
ducts, and, using the approxi
mations previously made, the model
of Fig. 432a may be reduced to
the simple representation of Fig.
4326. We might observe that
since there is no longer any trans
mission from the input to the
output, the slope of the transfer
characteristics will be zero.
Figure 433 shows the input
output transfer characteristics. In addition, the shift in axis produced
by the bias current h q is also accounted for by the second ordinate
(dashed line).
&n
Rz
• X
I
/ i
'w fit
h(t)
Fig. 433. Transfer characteristics of the
circuit of Fig. 429.
+20 v
+20 w
\ T 1
i V
1 1 —
I
i
r
a t.
(d)
Fig. 434. Circuit, models, and waveshapes for Example 42.
Example 42. The model techniques are also quite useful in evaluating the time
response of active circuits containing energystorage elements. To illustrate the
methods employed, we shall solve the simple circuit of Fig. 434a. The switch, which
is initially closed, will be opened at < — and closed after steady state is reached.
130 MODELS AND SHAPING [CHAP. 4
Before the switch is opened the large base current flow saturates the transistor.
This statement can be verified by assuming that it is true and calculating the current
flow. The model of interest reduces to that shown in Fig. 4346. By taking the
Thevenin equivalent of the circuit seen looking into the transistor,
ii(0_) = 2^0 = 50ma
From the activeregion model of Fig. 434c, the conditions for saturation are given by
fin. X 2 K + (0 + 1)4. X 2 K = 20 volts
or it, = 100 ii&
and this current is greatly exceeded.
Solving the circuit of Fig. 4346, the voltage at the emitter is
"• (0_) = 2K+VkU2K 20 " 15volts
After the switch is opened, the emitter starts decaying toward 10 volts, with the time
constant
t,MX2K = 2 msec
But the base current also decreases as C charges. When it falls to 100 /ia or when
»„ — 10.1 volts, the transistor becomes active. The elapsed time will be found
from the exponential charging equation
v.  10 + Se'"i
by substituting in the final value. Virtually the complete exponential is used, and
thus
h = 4n = 8 msec
After the circuit becomes active, the increase in input impedance to (fi + 1)2 K
[from Eq. (442)] increases the charging time constant to
t 2 = 1 /if(51 X 2 K + 1 K) = 103 msec
The discharge continues toward zero with this very long time constant. Finally
the switch is closed and the transistor switches back into saturation.
The waveshapes of the base current and the emitter voltage are shown in Fig. 434d.
46. Transistor Gates. If we reexamine the basic premise from which
we constructed the various circuit configurations used as gates, i.e., the
opening or closing of a transmission path upon signal excitation, it would
seem that perhaps a series arrangement of active elements would serve
as well as the parallel gates of Sec. 43. With triodes, the high plate
supply voltage necessary for their proper operation would have made
any such discussion academic. However, at this point we have at our
disposal the transistor, an active element requiring very low operating
voltages, and we may therefore contemplate this alternative configura
tion. Of course this does not prevent the use of transistor parallel gates.
The series gate fulfills its function as the individual transistors switch
from the almost direct short they present under saturation to an open
circuit when cut off. Since the opening of a series transmission path any
Sec. 46]
SIMPLE TBIODE AND TRANSISTOR CIRCUITS
131
where along its length is equally effective, for an ok gate (Fig. 435) the
individual transistors would normally be biased fullon. As any one
changes state into cutoff, the output path opens and the gate indicates
the presence of the input pulse. Conversely, in an and circuit (Fig. 436),
all the transistors would be biased below cutoff, with the output terminal
remaining above ground potential until the simultaneous excitation at all
e 3 oVW
t°
t°
e 2
t°
Ebb
t
Fig. 435. Series oe gate and waveshapes.
e,oJ\/vv
E bb
6 Ecc ^
Fig. 436. Series and gate and waveshapes.
inputs closes the gate. An important contrast between these two gates
is that the or gate, being biased on, requires heavy saturation current
flow under standby conditions, whereas the series and gate only con
ducts when its logic situation is satisfied.
One of the major problems facing the designer of logical systems, such
as digital computers and control equipment, is the power requirements
imposed by the large number of active elements employed. Besides
the expense involved in supplying the current drain at the regulated
voltages needed, the heat produced by the dissipated power must be
conducted away from the equipment. Otherwise the ambient tempera
132
MODELS AND SHAPING
[Chap. 4
ture of the enclosed components may rise to a point where some elements
will be seriously damaged or even destroyed. To this end, the engineer
seeks circuits operating with little or no standby power. Furthermore,
since a single triode requires approximately 1 watt of filament heating
power, transistors have almost completely superseded tubes in many
critical applications.
The two transistor gates which best satisfy the above specifications
are the parallelconnected ok gate and the seriesconnected and gate.
Each is normally biased below cutoff. Combinations of these two find
wide application in the development of systems of binary logic employed
B
fcu
k^T^
Fig. 437. Directcoupled complementary transistor gate producing an output upon
satisfaction of A or B and C.
in digitally operated equipment. Moreover, complementary design, i.e.,
use of alternating pnp and npn gates, permits the output of one gate
to be directly coupled as the input to the next gate.
Consider, for example, the circuit of Fig. 437, where all the transistors
are normally cut off: T A , Tb, and T c by having their bases returned
to the appropriate polarity bias voltage and To by adjusting the network
of Ri, Rd, and R E so that the voltage from point E to ground will be above
zero. Satisfaction of the parallel oh gate, upon the positive excitation
of either A or B, will force the related transistor into conduction, with
the voltage at point D now dropping to zero. The base of To becomes
negative, as a consequence of returning the coupling resistor Re to ground
through T A or Tb, and the series and gate is now primed for conduction.
As soon as a negative pulse appears at point C the gate opens, allowing
the output to rise from E ec to zero. If instead a negative output pulse is
Sec. 47] simple tbiode and transistor circuits 133
desired, the load could be shifted to the emitterground circuit of T c ,
with the output now dropping from zero to E cc upon the gate's closure.
We see that iHhe transistor is allowed to switch between cutoff and
saturation, then its simplified model representation is an open circuit
followed by a short circuit. Only in so far as the saturation value of
base current depends on the transistor parameters will the circuit's
operation be affected by the particular type of transistor chosen. Pro
vided that the threshold value is exceeded for the lowest value of
expected, the switching performance becomes completely independent
of the particular active element and would not deteriorate upon the
substitution of an entirely different transistor.
47. Pentodes. Both the transistor and the triode are threeterminal
devices, and therefore only two equations, involving four variables, were
necessary to define their complete operation. Moreover, the approxi
mations made in linearizing their characteristics were not so extreme as
to be incompatible with the actual physical functioning of the device. If
we attempt to extend this concept to the pentode, we find ourselves in
some difficulty. We are now forced to consider the related response of
four terminals instead of two (all voltages are specified with respect to
the cathode, the reference terminal). Each defining equation will
involve the terminal characteristics of the remaining three tube elements.
As a specific example, the equation of plate current must include its own
voltage as well as terms reflecting the effects of the suppressor, screen,
and controlgrid voltages on the plate current flow. These latter terms
appear as controlled sources in any model drawn.
h = /(eiAiAsAs) (445)
Analogous equations could be written for the other pentode elements.
First of all, in order to expand Eq. (445) into a form amenable to
expression as a piecewiselinear model, we would have to examine the
curves of plate current and voltage drawn with each of the three grid
voltages as a parameter. From these families of characteristics we
might be able to estimate the relative error introduced by linearization.
After replacing the actual curves by straight lines, the coefficients of the
linear equation are evaluated from the various slopes and intercepts.
Since the manufacturer does not usually make such data available, the
engineer would have to perform his own measurements. As the final step,
one or more biased diodes would be inserted to restrict the range of
operation at the plate.
Assuming that reasonably satisfactory linear operation is possible, Eq.
(445) may be expanded so that it leads to a currentsource representation.
U = ff,A + guei + j/acj + gufit (446)
134
MODELS AND SHAPING
[Chap. 4
The numerical subscripts indicate the particular grid under consideration.
Each of the coefficients represents the transfer admittance from the
appropriate grid to the plate. If we restrict operation to the first
quadrant, the model representation
of the plate circuit is as shown in
Fig. 438.
A similar process carried out for
each element of the pentode will
result in similar models. In solv
ing any specific problem, all four
models would have to be solved
simultaneously and the answers
individually checked to ensure that
Fio. 438. Model representation for the
plate circuit of a pentode.
none of the boundary values are exceeded.
It appears that the desire for generality in this case has greatly increased
the complexity, even to the point of allowing the model to obscure the
actual physical processes. Therefore we might best deal directly with the
characteristics rather than bother to construct a model.
In many specific circuits, however, the tube voltages are so restricted
that simpler models may be drawn. The suppressor primarily serves
to determine the division of the cathode current between the plate and
the screen grid, while the total
cathode current depends on the
screengrid voltage. If both of
these are held constant, then
the plate voltampere character
istics are almost constantcurrent
curves, with the particular value
of current determined solely by
the controlgrid voltage (Fig.
439). The model for the plate
circuit operation would be reduced
by combining the two constant
terms of Fig. 438, gy,ei and g 3 be 3 ,
into a constantcurrent generator
/„ equal to the current intercept
of the zero gridvoltage line.
Each of the pentode's grids has sufficient control over the plate current
flow so that if any one or all are made highly negative, the tube is
completely cut off. We can thus utilize any individual grid as a gate
input, and for special gating functions we may employ various combina
tions of the three grids. They function in series as valves. Changing
any one from the on to the off state gives us the or gate. If all elements
20 ma
H
h
£<*
2
. 4
lOma
6
8
10
_ 12
lOOv 200 300 c »
Fig. 439. Pentodeplatecircuit voltam
pere characteristics — constant screen and
suppressor voltage.
Sec. 48]
SIMPLE TBIODE AND TRANSISTOR CIRCUITS
135
are normally biased off, they must all be turned on before plate current
will flow (and gate). Another commonly used pentode circuit is the
controlled gate of Fig. 440. Here the suppressor will maintain the plate
cutoff until the application of a control pulse. Once current flows, the
tube operates in a normal manner, amplifying and transmitting the con
trolgrid signal to the plate circuit.
We note that advantage is taken of the control characteristics in much
the same manner as the series arrangement of transistors. Each control
element, reflected as a controlled source, has the ability to turn the device
on and off, and by acting in unison, they create the logical situation
sought.
t
"•"We
^A
Fig. 440. Pentodecontrolled gate.
48. Summary. The importance of the model concept in freeing the
engineering viewpoint from the conformity imposed by the rigor of
graphically presented data cannot be overstressed. It is extremely
difficult to give the imagination free reign when the active element, the
heart of electronics circuit design, must be treated in a manner different
from its associated circuit components. Constant referral back and forth
from the tube or transistor graphics to the circuit equations involving
the passive elements denies the grasp of the basic nature of the system
behavior; it leads to the treatment of each circuit as a separate entity.
In direct contrast to this approach, the very nature of the simpli
fying approximations made when drawing a model must lead to a simpler
presentation of the phenomena under examination. Furthermore, the
essential unity of circuits and systems immediately becomes clear from
the similarity of their models. Regardless of the type of control, be it
voltage, current, pressure, velocity, or temperature, an equivalent con
trolledsource representation places the controlling element in the fore
136 MODELS AND SHAPING [CHAP. 4
front. After the piecewiselinear model of the nonlinear element is drawn
and the remaining circuit components are placed in their proper relation
ship, then, within each region, analysis proceeds as in a linear circuit.
All linearcircuit theory becomes applicable, and many methods that
would not normally be applied give power to the engineer.
PROBLEMS
41. (a) Construct a piecewiselinear model to represent the 12AU7 which will
agree exactly with the tube characteristics at E c = — 10 and h = 15 ma. Give all
parameter values.
(6) Superimpose your model on the tube characteristics of Fig. 41 and delineate
the regions where the agreement is within 20 per cent of the actual current.
(c) Construct an alternative model which will hold for currents less than 5 ma.
How does it differ from the one found in part a?
42. (a) Drawing on the answers of Prob. 41, construct a model which will repre
sent the plate characteristics by two contiguous segments. In this manner we can
obtain closer agreement with the measured curves.
(6) For this model repeat part 6 of Prob. 41.
43. The 12AU7 is used as a simple plateloaded cathode bias amplifier with Rl =
12 K and En = 250 volts. Calculate the incremental gain as found from the piece
wise model and from the actual characteristics at the following bias voltages: +5, 0,
—5, —10, and —15 volts. Express the deviation from the value found from the
model as a percentage. (Assume that r„ = 1,000 ohms in the positive grid region and
that the signal source impedance is 2,000 ohms.)
44. If we restrict operation to et > ke c (as shown in Fig. 42), show that the
approximate platecircuit model for e& < ke c is simply a resistor of kr p /(fi + it).
46. Calculate and plot the transfer characteristic for the circuit of Fig. 45a when
the cathode bypass capacitor is removed. The tube parameters are r p = 70 K,
It ■> 100, and r c = 1 K, and we choose a load resistor of 200 K with E b b ■» 250 volts.
What value must we specify for R t if the quiescent current is to be 0.5 ma?
Sketch the output if the input is a triangular wave of 50 volts peak to peak.
46. The three circuits of Fig. 441 illustrate the most common amplifiers. The
same tube and load resistor are used in each case (r„ =■ 25 K, p — 50, Ri = 50 K, and
r.  1 K).
SIMPLE TRIODE AND TBANSI8T0B CIKCUITS
137
(a) Draw the complete transfer characteristics of all three circuits on the same axis.
(6) Specify the output impedance of each amplifier in each of the three regions of
operation.
(c) Specify the input impedance of each circuit in each region.
(d) Can you draw any conclusions as to the applicability of these amplifiers?
47. (a) Calculate and plot e„ versus time for the circuit of Fig. 442 when C =
200 «if and R = 25 K. The input signal is periodic.
(6) Repeat part a with C  2,000 ntf and R  250 K.
(c) Repeat part 6 when R is connected between the grid of 7/j and 300 volts instead
of being returned to ground.
+300 v
«*
70
*300_
/»sec
200
/jsec
t
Tjand T 2 : 12AU7
Fig. 442
48. At what time after the switch is opened will the tube in Fig. 443 start con
ducting and how much later will the grid conduct? (p — 20, r T = 10 K, and r, —
1 K.) Sketch and label the waveshapes appearing at the plate and grid.
+ 300v
Fig. 443
49. Repeat Prob. 48 when the RC combination in the grid circuit is replaced by
a 10K resistor and a 100mh inductance, respectively.
410. In the circuit, of Fig. 411, n = 70, r, = 50 K, Eu,  300 volts, «,  100 K,
and Ea " — 150 volts. Find the value of JR. that will ensure symmetrical clipping of
the input signal. Plot and label the transfer characteristics.
411. Consider the cathodecoupled clipper of Fig. 411 having the following
parameters:
r,  70 K ft,  100 K Eu  300 volts
100
«•
JE«.
150 volts
138
MODELS AND SHAPING
[Chap. 4
(o) Calculate the value of R, that wOl result in a quiescent current of 0.6 ma
through T t . (Hint: Since the commoncathode voltage is approximately zero, the
drop across R t and T 2 is almost exactly equal to En.)
(b) Calculate the two clipping levels with respect to the input.
(c) Sketch and label the transfer characteristic.
412. In the circuit of Fig. 441a (Prob. 46) a triangular wave is coupled through
a 0.01iii capacitor rather than the 10K resistor. What is the largest possible peak
topeak input signal before the circuit begins to clip the input? Sketch the output
when the input is twice as large as found above.
413. (a) A cascode amplifier, such as shown in Fig. 444, is widely employed in
television sets where the noise must be kept within bounds. If both of the triodes
shown are 12AU7's, calculate the quiescent operating point and the transfer character
istics. Draw the simplified models holding at each break point.
(6) Explain the steps which must be taken in order to find the quiescent point by
a graphical construction. Check the answer to part a by this means.
Fia. 444
414. (a) Calculate the value of .fti that will set the quiescent output of the circuit
of Fig. 445 at zero (E, = E cc = 150 volts).
(6) What kind of gate is this? What is the minimum input amplitude that will
produce the maximum output amplitude?
416. In the circuit of Fig. 445, Ei is set equal to E hb . The load resistor R t is 50 K,
and it is shunted by 150 nni stray capacity.
(a) Sketch the output if we apply — 100volt 20msec pulses to ei at t = 0; to e 2 ,
5 msec later; and to e 3 at t = 10 msec.
(6) Repeat part o if the inputs are changed to 20volt positive pulses.
(c) Can you see any way to improve the operation of this gate?
416. Repeat Prob. 415 when the load resistor is transferred to the plate circuit
making the and gate of Fig. 416. The cathodes are returned to ground.
417. We wish to eliminate the pedestal present in the controlled gate of Fig. 417.
The tube and circuit parameters are r p = 100 K, p = 100, Rl = 200 K, and JBu, =
300 volts. The gating tube T 2 is normally maintained cut off by the 12volt dc level
of the gating signal applied at the grid of Ti; superimposed 10volt negative pulses
turn Ti on.
SIMPLE TKIODE AND TRANSISTOR CIRCUITS
139
Use a tube with the same parameters and calculate the value of the cathode resistor
necessary to completely eliminate the pedestal. Assume that the control pulses are
coupled into its grid through a large capacitor. Specify the amplitude and polarity of
the signal which must be simultaneously applied to this additional triode.
?200v
0,10 K
lt20
r c lK
418. There are four possible models which may be used to represent the four
variables it, tij,,, i c , v„ in the groundedemitter configuration. Draw these models and
show where diodes must be inserted to restrict the operation to the proper quadrant.
Give the meaning of the various circuit parameters in terms of those given in the
model of Fig. 427.
419. The circuit of Fig. 446a is used as a currentvoltage converter.
(a) Plot the transfer characteristics (e* versus ij), taking all terms into account,
and compare it with the plot found from the appropriate approximate models.
(6) Calculate the input admittance and output impedance in each of the three
operating regions.
(c) Plot the output if i'i = 50 sin at ma.
Fig. 446
420. We wish to choose Ri, in the circuit of Fig. 4466, so as to achieve symmetrical
clipping. Under these circumstances repeat Prob. 419.
421. The transistor used in Fig. 446 has an I c „ of 10 ix& at room temperature, and
it doubles for every 10°C rise. If the value of R\ in Fig. 4466 is 100 K, by how much
will the Q point shift with a 20°C rise in ambient temperature? Repeat this calcula
tion for the circuit of Fig. 446o and express both shifts as a ratio of the Q point
calculated when I r .o is assumed zero.
140
MODELS AND SHAPING
[Chap. 4
422. The circuit of Fig. 447 employs transistors having r.  30, n  300, and
= 20. Sketch to scale the input e t and the output e, when the excitation shown is
applied. Two cases should be considered:
(o) When C = 1 /if.
(6) When C  0.001 /if.
9+10 v
«i
10 v
100
/*sec
Fio. 447
428. Sketch and label the collector voltage in the circuit of Kg. 448 if the switch
is opened at t «■ and closed after the transistor is forced as far into saturation as
possible. Make all reasonable approximations and indicate the times at which the
transistor changes state.
?+20v
Fio. 448
424. As the reverse bias of a transistor drops to zero, the current gain of the tran
sistor also decreases. Figure 449 shows a semiquantitative view of this decrease for
the G.E. 2N123 pnp transistor.
(a) Using Fig. 449 and given the data that r, = 28 ohms and n = 80 ohms, find
the input impedance of the transistor when it is biased such that e„ = 0.
t bu
] 40
e 30
20
10,
s — ■
V
*§
lv
2
«c
Fig. 449. The transistor parameters are n = 200, r. = 20, r« => 2 megohms, and
a  0.95.
SIMPLE TBIODE AND TRANSISTOR CIRCUITS
141
(6) The maximum emitter current ia to be 1 ma. What values of Ri and Rl will
just meet this requirement?
(c) What value of ei is necessary to just cut off the transistor?
425. The circuit of Fig. 446o is modified by connecting a 1henry choke from the
emitter to ground. The input is a 10kc current sine wave of adjustable amplitude.
If the signal distortion due to the periodic charge and discharge of the coil is neglected,
what is the peak input amplitude before the output appears clipped?
426. A parallel combination of transistors is connected as an and gate similar to
the one shown for the triodes in Fig. 416. Their collectors are connected together,
and the output load resistor inserted in this circuit. We desire to drive this gate with
negative pulses at the base.
(o) If we have both npn and pnp transistors, a 2,000ohm load resistor, and two
20volt batteries at our disposal, what circuit would be used?
(6) How large must the input switching pulses be for the maximum output swing?
The voltage pulses are applied through 20K resistors directly into the base. Any
bias resistors selected must furnish a base current of five times the saturation value.
(c) Repeat this problem if the pulses are injected into the base and the output load
is connected from the common emitter to ground. Transistor parameters are
t% = 150 ohms
a  0.98
r, = 30 ohms
r„ — 2 megohms
Make all reasonable approximations to simplify your calculations,
427. Using either npn or pnp transistors having the parameters given in Prob.
426, we desire to construct a parallel or gate. It should operate on positive signals
injected at each emitter and should furnish an output across a 2,000ohm resistor in
the commonbase circuit as well as across a 1,000ohm resistor in the commoncollector
circuit.
(o) Draw a threetransistor circuit when two 25volt batteries are available for
power.
(6) If the bias resistors are 2 K, what is the impedance seen by the driving signal
source?
(c) The input signal at one emitter is 10 per cent above the value necessary to
drive the transistor into saturation. Specify this signal and draw both outputs to
scale.
428. In the gates of Figs. 435 and 436 the transistors used are those given in
Prob. 426. In both cases Rl  2 K, R t = Ri = R, = 20 K, E» = 20 volts, and
Ece  20 volts.
+•»
3v
&msec
4v
i e 2
2 4
t, msec
Fig. 450
«3
::2i
OT 15"
fcmsec
(a) We apply the signals given in Fig. 450 to these inputs through a 2,000ohm
source impedance. Sketch to scale the resultant outputs. (Assume that the signal
142
MODELS AND SHAPING
[Chap. 4
polarity is inverted for the ok gate and make all reasonable approximations in your
calculations.)
(6) Calculate the input impedance seen at es, in both gates, under each condition
of operation.
429. Figure 451 shows an npn gate directly coupled to a pnp gate. When no
pulse is applied to the first gate, both gates are "on," that is, v e = 0. When, however,
a negative pulse appears at the input of T u both gates are cut off.
(o) What is the approximate input impedance of each transistor?
(6) When €i = 1.5 volts both transistors are just cut off, and when ei = both
are just saturated. What values of R lt Ri, R lt and R, are necessary to produce this
effect? Make all necessary valid assumptions to simplify the analysis.
?+1.5v
1.6 w
r 6 =200
r e =40
„ /3foc«OJ9
Fig. 451
r„=80
r e =28
/Sd^OlO
430. The circuit of Fig. 452 is adjusted so that each transistor is normally biased
cut off. Sketch the output waveshapes at all three terminals in proper time sequence,
specifying all voltage values.
f+4v
e l
_4v
10 K
10
5 15
10 K
10 K
Fio. 452
431. The two transistors in the gate of Fig. 453 are characterized by the two
curves shown. They hold when the collector is backbiased, but when it is forward
biased, we can take v„ = 0.
SIMPLE TRIODE AND TRANSISTOB CIRCUITS
143
(a) If the excitations are as specified, sketch the outputs at e 2 and e> to scale, giving
all values.
(6) Repeat part a if the collector supply voltage is changed to —3 volts.
t
300 mv
+ 600 mv
t
150mv
300mv
12 3 4 5 6 7
t,
msec
12 3 4 5
— L_l
6 7 i
msec
200
te
30
ma
va, mv 200 300
Fig. 453
vtt» mv
482. Design a pentode and circuit such that there will be an output if two inputs
are positive and there will be no output if either of the two inputs or both inputs are
zero. Draw waveforms showing all these possibilities. Specify the inputs and out
puts. Numerical answers are not required, but the relative values should be stated,
i.e., which voltages are negative and which are positive and whether one voltage should
be much higher than another for proper operation.
BIBLIOGRAPHY
Angelo, E. J., Jr.: "Electronic Circuits," McGrawHill Book Company, Inc., New
York, 1958.
Clarke, K. K., and M. V. Joyce: "Transistor Circuit Analysis," Addison Wesley
Publishing Company, Reading, Mass., in press.
Ebers, J. J., and J. L. Moll: Large Signal Behavior of Junction Transistor, Proc. IRE,
vol. 42, no. 12, pp. 17611772, 1954.
Lo, A. W., et al.: "Transistor Electronics," PrenticeHall, Inc., Englewood Cliffs,
N.J., 1955.
Middlebrook, R. D.: "An Introduction to Junction Transistor Theory," John Wiley
& Sons, Inc., New York, 1957.
Shea, R. F.: "Transistor Circuit Engineering," John Wiley & Sons, Inc., New York,
1957.
Zimmermann, H. J., and S. J. Mason: "Electronic Circuit Theory," John Wiley &
Sons, Inc., New York, 1959.
PART 2
TIMING
CHAPTER 5
SIMPLE VOLTAGE SWEEPS, LINEARITY,
AND SYNCHRONIZATION
The preceding chapters were concerned with the shaping of voltage
and current signals by various configurations of linear and nonlinear
elements, e.g., differentiation, integration, amplification, clamping, clip
ping, and gating. No consideration was given to the origin of the
signals applied, but as we now begin this study, we should realize that
many of the analytic techniques previously developed are applicable to
our paramount problem, the generation of simple waveshapes.
In this chapter we shall concentrate our attention oh some character
istics of one particular waveshape, the linear voltage sweep. Ideally, the
output voltage increases linearly with time until it reaches a predeter
mined final value, instantaneously returns to zero, and immediately starts
increasing again as the cycle repeats. This waveshape finds widespread
application: in the horizontal deflection circuit of an oscilloscope so
that timevarying signals may be displayed against a linear time scale;
in radar systems, for measuring the time required for the return of the
echo signal; in television transmitters and receivers as a means of gen
erating the raster; and in various control systems where it is used to time
preprogrammed functions.
51. Basic Voltage Sweep. All linearvoltagesweep circuits, regard
less of individual variations, have a common basic mode of operation:
some form of exponential charging; an automatic change of circuit which
introduces a discharge path at the proper point of the cycle; and then
reestablishment of the charge path. Differences in sweeps usually con
cern the active elements controlling the points at which discharge com
mences and ceases. But to approximate more closely the ideal sweep,
we are sometimes also forced to modify the charge and discharge paths.
Thus the basic voltage sweep might well be characterized by the circuit
shown in Fig. 5la, which generates the waveshape of Fig. 516. The
capacitor charges through Rx toward E^. When its voltage reaches some
preset final value E f , we throw the switch to position 2, discharging C
through Ri. Upon decaying to its initial value E i} the switch is returned
to position 1 and the cycle repeats.
147
148
TIMING
[Chap. 5
Before turning to practical sweep circuits, we can reach some con
clusions as to their general requirements from a closer examination of
Fig. 5la and 6. Of prime importance is the recognition that we are
generating, not a linear sweep, but an exponential charging curve, and
only through restricting operation to a relatively small portion of the
total possible curve will the capacitor voltage even approximate a linear
rise. Thus E» must be made much higher than the desired output swing
E r Ei to ensure reasonable linearity. Secondly, we note that the retrace
time, a direct function of Rz, becomes insignificant only when the dis
charge time constant is very small. Making Ri zero obviously reduces
the discharge time to zero, but other circuit considerations may negate
this choice. Therefore R2 should be chosen as small as possible, con
sistent with any limitations imposed.
Ebb
^_i::_™."_
kE,
>o(t)
Ei
/
y
v\
! ! ^_
< —
T,—
Xr 2 J '
Fig. 51. (a) Basic voltage sweep circuit; (b) output sweep voltage.
Finally, we can write the equation of the capacitor voltage during
either portion of the cycle and, by substituting the known final value E/,
solve for the sweep duration. During the voltage rise, when n = R X C,
Thus
Tx = ti In
E»  (Etb  Ede'i'i
En, — Ei
(51)
Em — Ef
The capacitor discharges from Ef toward zero with a time constant
r 2 = R2C, and consequently the sweep recovery time, the time required
to fall to Ei, becomes
(52)
r s = r 2 lnfj
The only remaining unknown is the mechanism of circuit switching,
which must be evaluated in terms of the physical characteristics of the
particular device employed.
52. Gastube Sweep. An extremely simple sweepvoltage generator
(Fig. 526) makes use of the doublevalued characteristics of a cold
cathode gas tube to periodically discharge the sweep capacitor. As
Sec. 52]
SIMPLE SWEEPS AND SYNCHRONIZATION
149
long as the anode voltage remains below the ignition potential, E f of
Fig. 52o, it supports only a very limited leakage current flow; for all
practical purposes the tube is an open circuit. However, once the
voltage rises to E f and the tube fires, the terminal voltage drops to a lower,
almost constant value E . Both E f and E Q depend on the particular gas
used. The tube current is now primarily limited by the external circuit
and C begins discharging through the path established by the ionized gas.
The discharge path will be maintained only as long as the current
flow is sufficient to sustain ionization. As C discharges, the tube current
drops, tracing the dashed path of Fig. 52a instead of retracing the orig
inal curve. Because we are interested in generating a repetitive wave
shape, we must ensure the existence of circuit conditions that will
extinguish the tube, preferably rapidly.
\" /
J? 2
— — — — — 1 o
(a) (b)
Fig. 52. (a) Gastube voltampere characteristics; (6) gastube sweep circuit.
The mechanism of the gasdischarge extinction, the recombination
of ions previously produced by the large forwardcurrent flow, requires
some small finite time. It is impossible to state the deionization time
in simple terms, since it depends on the past history of the tube, i.e., on
the peak current on ignition and on the time rate of current change
during the capacitor discharge. This time varies from tube type to
tube type and even changes in any given tube during its life; it usually
ranges between 0.5 and 2 /usee.
In order to simplify the calculations, we find it convenient to assume
that the tube voltage remains unchanged during the whole discharge
interyal. Additionally, a fictitious extinction current I bx might well be
postulated as a means of representing the complete, complex extinction
phenomena. Thus the following statement suffices: when the tube
current drops below I bx , the tube goes out. I bx depends on the tube
type in much the same manner as does the extinction time, with its value
generally falling between 25 and 250 n&.
During capacitor charging, the nonconducting tube can be omitted,
leaving an equivalent circuit identical with that of Fig. 5la (switch
150
TIMING
[Chap. 5
in position 1). Therefore the rise time is identically given by Eq. (51).
However, evaluation of this equation is contingent on finding the still
unknown initial value E t . Referring to the output waveshape (Fig. 536),
we see that the initial value of the rise corresponds to the final value of the
decay, i.e., to the point at which the tube extinguishes.
■ — vw
*>A/\ — I
=£„ ?fcc =rE,
(a)
Fig. 53. (a) Discharge circuit of gastube sweep; (b) gastube sweep waveshape.
The circuit enters the region of operation defined by Fig. 53o with the
capacitor voltage at E/ and leaves it when the output drops to E,. Imme
diately upon the tube's firing, C starts discharging toward the Th6venin
equivalent voltage across its terminals.
Et =
Ri
Tti  X12
E +
/L2
Ri + R
Eu
(53)
Extinction occurs when the tube current falls to h x , which corresponds
to a capacitor voltage of
e c (T 2 ) = Ei = E + I hx Rt
(54)
Since we now know the initial, final, and steadystate voltages in this
region, the decay time becomes
Et
E,
where
Et — (E + IbxRi)
r 2 = (fii  R*)C ^ RiC
(55)
Normally, R2 serves to limit the maximum tube current to a safe
value and is very much smaller than Ri. Reexamination of Eq. (53)
indicates that under these circumstances the Thevenin equivalent voltage
will be only slightly larger than E„. The second term of Eq. (54),
denning the initial voltage Ei, will generally be almost negligible compared
with the first term as a consequence of the small values of both h x and Ri.
E may vary from 10 to 50 or 100 volts, depending on the tube type, but
Sec. 52]
SIMPLE SWEEPS AND SYNCHRONIZATION
151
IbzRi is usually somewhere between 25 and 250 mv. The steadystate
discharge voltage, the initial voltage, and the tube's extinction voltage
are so close together that we might as well approximate them by the
single value E . We conclude that the discharge curve will constitute
virtually the complete exponential, requiring roughly four time constants
for completion.
T 2 ^ 4iJ 2 C (56)
Occasionally, the deionization time may be longer than the time given by
Eq. (56), and if it is, it will predominate.
A glance at Fig. 53<x discloses that during ignition part of the tube
current is contributed by Ebb. If this component is greater than I bt ,
the tube will never extinguish. To prevent continuous ignition, the
inequality of Eq. (57) must be satisfied.
h = RT+R* < h * (5 " 7)
Equation (57) sets the lower limit of R x necessary for guaranteed repeti
tive sweep generation. Since Ri >5> Rz,
Ebb — E
Ri >
(58)
In practice, the circuit may operate properly when Ki is from onehalf
to onequarter of the limiting value given by Eq. (58). This apparent
inconsistency arises because h x is only a convenient fiction approximating
the whole discharge phenomenon
and a steadystate current several
times as large may still allow
extinction.
Limits must also be imposed on
Ri\ if too large, the discharge time
becomes excessive, and if too small,
the peak current on ignition may
exceed the tube's rating. There
fore it must satisfy
R2 >
Ef — Eg
(59)
Fig. 54. Avalanche diode characteristic.
where Ib, ml is the maximum allow
able tube current.
Several semiconductor devices have twoterminal voltampere char
acteristics quite similar to that of the gas tube. Among these are the
avalanche diode, the unijunction diode, and the pnpn transistor. One
typical characteristic is shown in Fig. 54, and it should be compared
152
TIMING
[Chap. 5
with Fig. 52a. A somewhat more detailed discussion of these devices
will be deferred until Chap. 11, at which point some simple sweeps will
again be treated, but from a different viewpoint from that expressed
here. However, it is important to note that similar voltampere charac
teristics mean similar circuit behavior and therefore these semiconductor
devices may be used to replace the gas tube in the sweep of Fig. 526,
especially in lowvoltage applications.
63. Thyratron Sweep Circuits. The sweep of Sec. 52 has many
inherent disadvantages: among them, that its sweep amplitude is small in
comparison with the ignition potential and depends solely on the non
controllable firing point and sustaining voltages; that synchronization
cannot be effected through injection of external signals; and finally that
the relatively long recovery time precludes use for generation of short
duration sweeps. All but the last of these drawbacks are overcome in
one practical sweep (Fig. 55) by employing a thyratron in place of the
Fia. 55. Thyratron sweep circuit.
coldcathode gas tube. Except for the control afforded by the grid, this
circuit performs identically with the sweeps discussed in Sees. 51 and 52,
and consequently all equations previously derived are applicable.
Where one curve sufficed to define completely the operation of the
coldcathode gas tube, two are required for the thyratron because of its
additional control element: one expressing the gridvoltagefiringpoint
relationship (Fig. 56a), and the other, the plate voltampere character
istic (Fig. 56&). The thyratron grid is a massive structure, physically
situated so that it almost completely shields the cathode from the
influence of plate voltage. A negative grid voltage establishes a potential
barrier, preventing electron travel from the cathode to the plate, cutting
off the tube. As the plate voltage increases, its influence on the potential
distribution in the interelectrode space between grid and cathode increases
proportionately. Eventually the potential barrier is lowered enough to
allow highenergy electrons into the gridplate region, where they collide
with gas molecules, ionizing them. This value of plate voltage is called
the firing point of the tube. Referring to Fig. 56a, we see that the more
negative the grid, the higher the plate voltage necessary for ignition.
Sec. 53]
SIMPLE SWEEPS AND SYNCHRONIZATION
153
Upon firing, the plate voltage drops to E , and as in the coldcathode
gas tube, the current increases until limited by the external circuit.
Since the drop across the thyratron is relatively small (10 to 25 volts)
in comparison with the firing voltage, it generates a largeamplitude
sweep.
Once the tube ignites, the positive ions are attracted by the most
negative element in the tube, the grid. They form a sheath on the
grid structure, effectively insulating it from the tube and preventing
further control. The grid current that now flows is limited to a safe
value by choice of the external resistor R„. Breaking the plate con
nection or reducing the tube current below h x extinguishes the tube, with
the grid now regaining control.
15<T
(b)
Fig. 56. (a) Thyratron firing and (6) thyratron plate characteristics.
Direct algebraic solution of thyratron circuits necessitates the algebraic
expression of the plate firingpoint characteristics. The simplest
representation is the dashed line shown in Fig. 56a. Its equation is
E, = mE c + Et &£ 9E C + 5 (510)
where E c < and E/ > E a . In this equation m is the slope and E
the intercept. Substituting Eq. (510) into Eq. (51), the sweep time
becomes
En, — E
Ti = ti In
(511)
Eu  (mE c + Ei)
With a supply of 250 volts and a bias of — 10 volts, the sweep lasts for
Ti = Tiln io^"§§ = a415Ti sec
Under the above conditions, the total sweep amplitude is 80 volts
(Ef — E t ) out of a maximum possible swing of 235 volts. Even over
154
TIMING
[Chap. 5
this large fraction of the charging curve, the sweep approximates a
straight line reasonably well.
The sweep period, a linear function of the time constant R\C, is varied
by changing C in steps for coarse control and by changing R\ continu
ously as a fine adjustment. In each range the minimum resistance must
satisfy Eq. (58), and therefore R\ usually consists of a fixed resistor
of the minimum value in series with a potentiometer.
Minor modification of the freerunning sweep of Fig. 55 converts
it into a singleshot generator (Fig. 57), which produces a single sweep
upon each application of an input trigger. In its normal state the plate
voltage is limited to E b by the plate catching diode D x . E c sets the firing
voltage well above this value. The injection of a positive trigger at the
grid momentarily lowers the ignition potential below Eb, and the tube
immediately fires. C now discharges, the tube extinguishes, and C
(a) "» (6)
Fig. 57. Singleshot sweep generator and output waveshape.
starts recharging, only now generating the single sweep shown. Because
sweep starts after the discharge is complete, unless the delay introduced
is small, a portion of the signal we wish to observe may not appear on the
oscilloscope.
54. Sweep Linearity. Since our announced objective is the gener
ation of a linear sweep, we should have a means of expressing the linear
ity, or departure from linearity, as a measure of the sweep quality. This
entails making a comparison between an ideal and the actual sweep,
with any divergence representing the nonlinearity. Three possible
choices are presented in Fig. 58. The first is a straight line drawn tan
gent to the sweep at the origin (Fig. 58o) ; the second is a compromise,
intersecting the sweep at the point that results in equal deviation above
and below the line (Fig. 586) ; the third line connects the end points of the
sweep (Fig. 58c).
Depending on the particular application of the sweep, any one of the
three will be eminently satisfactory as a basis for comparison. When
ever the sweep time duration becomes important, the sweep is usually
synchronized by an external reference signal which constrains its end
Sec. 54]
SIMPLE SWEEPS AND SYNCHRONIZATION
155
points (Sec. 55). Under these conditions it seems reasonable to calculate
the amplitude deviation from an ideal sweep connecting the two restricted
ends. Referring to Fig. 58c, the deviation is
«(*) = e„(0  «,(*)
(512)
The ratio of maximum deviation to sweep amplitude, expressed as a
percentage, is defined as the sweep nonlinearity.
NL = ^ 100%
(513)
where E. = E t — E { .
Fig. 68. Various methods of defining Bweep nonlinearity.
By shifting the axis and writing all equations with respect to E it the
equation of the linear sweep becomes
e t (t) = kt
and that of the exponential sweep
E..
«.(*) = (En  E t ){X  e"")
(514)
(515)
When the limits of E, and h are substituted into Eq. (515), the coefficient
can be expressed in terms of the known sweep quantities.
Eti
E t 
E.
1 _ etjrt
(516)
The first two terms of the powerseries expansion of the exponential
[Eq. (517)] represent its linear approximation, and the remaining higher
order terms, the deviation from linearity. Because we are interested
in an almost linear sweep, the contribution of the higherorder terms
must be kept to a minimum, though using only a small portion of the
total exponential curve. Therefore t/r\ will always be much less than
e '/'. = i _
ri
156 timing [Chap. 5
unity and we can assume that the complete nonlinearity is due to the
square term of Eq. (517).
Substituting Eq. (516) into (515) and using the first three terms
of the expansion in both numerator and denominator, the equation of the
exponential sweep becomes
(A ~ TlE ' \L _ I ^lYl
6cW = «,(1  h/2r0 [r» 2 \ T1 ) J
and since ti/Vi <K 1 ,
The one missing piece of information, the time location of S m . t , will
be found from geometric considerations. Any curve having only slight
curvature, such as the exponential sweep of Fig. 58c, may be approxi
mated by the arc of some circle with large radius. The linear sweep
becomes a chord of this circle, and the maximum distance from a chord
to the arc, measured at any fixed angle, occurs at the center of the chord,
at the point corresponding to ti/2. We evaluate Eqs. (514) and (518)
at this time and substitute the answers into Eq. (512), with the result
«• GH£A®']*
^~E, (519)
OTl
Consequently, the percentage nonlinearity becomes
NL = ~ 100 % (520)
OTl
Substitution of the exact period, expressed in terms of the circuit voltages
[Eq. (51)], provides us with an alternative form for the nonlinearity,
one which brings to the forefront the dependency on the voltage limits
imposed.
A third expression for the nonlinearity is possible when the sweep
starts close to zero and constitutes only a small fraction of the total
exponential If E» » #, and En » E s , then by expanding Eq. (521)
Sec. 55] simple sweeps and stncheonization 157
into a series and taking only the first term,
NL£^12.5%Ji (522)
Equations (520) to (522) verify our previous contention that good
sweep linearity can be realized only by operating over a small portion
of the total exponential. For less than 1 per cent nonlinearity, the sweep
time must be restricted to 8 per cent of the time constant and its amplitude
to 8 per cent of possible charging voltage.
Since the other possibilities presented in Fig. 58 referred to the same
sweep, with the same amplitude, end points, and time constant, any
nonlinearity defined in their terms will differ from Eq. (520) or (521) only
by a constant multiplier.
55. Synchronization. The period of a sweep depends on many fac
tors: on the circuit time constant, on the voltage at which the capacitor
discharge commences and at which it terminates, and finally, on the
steadystate supply voltage. These parameters are never completely
constant but vary slightly because of ambienttemperature change, line
voltage variation, aging of tubes and components, noise, etc. As a
direct consequence of the random circuit changes, sequential sweep cycles
will not have identical duration but will jitter about some average time.
When this sweep is used for oscilloscope display, a corresponding jitter
appears in the pattern seen on the screen. If used for timing, there will
be uncertainty as to its exact time duration. However, by injecting an
external control signal, the sweep time can be locked to some multiple
of the synchronizing signal period, and when this input is derived from
the display signal, the oscilloscope pattern will remain stationary.
Synchronization of the sweep is accomplished by forcing the free
running sweep period h to some integral multiple of the signal period
t„ that is, by changing h to nt,. In effecting the change of period, the
sweep amplitude may also be affected, but as the amplitude can always
be adjusted elsewhere in the system, this is relatively unimportant.
The sweep period is controlled by varying either the initial voltage
or the discharge point. For example, in the thyratron sweep of Fig. 55,
the sync signal is injected at the grid, with its fluctuation about the
quiescent grid point changing the instantaneous firing voltage. Since
the slope of the plate firing characteristic [Eq. (510)] represents the
"gain" of the gas tube, the effective variation at the plate will be m
times as large as the applied grid signal. Figure 59 indicates the firing
voltage in the presence of a sinusoidal sync signal. We observe that,
because of the assumed linearity of the gridplate transfer characteristics,
the grid waveshape is simply multiplied, reflected into the plate, and
plotted about the quiescent firing line. The sweep ends earlier because
158 timing [Chap. 5
it intersects the firing curve at a lower voltage value. As a consequence
it fires at the same point of each cycle, whereas the freerunning sweep
(dashed line of Fig. 59) terminates randomly with respect to the applied
sinusoidal signal.
If we use an ideal freerunning sweep instead of an actual thyratron
sweep, the quantitative treatment below will be greatly simplified. In
addition, the lack of identification with any particular circuit increases
Csync
Fig. 59. Applied synchronizing signal and its effect on sweep duration.
Ey+Es
E f
E f E.
hvH
*X
•Esync
/ if v f \
1 /
1 /
°<
D
1 /
1 /
1 /
h
« 2t, H
t
Fig. 510. Ideal sweep synchronized by a triangular wave.
the generality of our discussion. This perfectly linear sweep starts rising
from zero at a rate of k volts/sec, ending at the firing voltage h sec later.
The introduction of a sync signal of proper amplitude and period
pulls the sweep into synchronization as shown in Fig. 510. This buildup
may take one or several cycles, but eventually the sweep will start and
end at the same point of the sync cycle.
One obvious question facing us is, Why does synchronization increase
the stability of the sweep with respect to random circuit changes? Our
argument might begin with the observation that these changes result in
sweep time instability, and since the effect produced is the same, any
circuit variation may be represented as an equivalent small perturbation
Sec. 55]
SIMPLE SWEEPS AND SYNCHRONIZATION
159
of firing voltage. In Fig. 51 la, the slight increase of Ef by AEf increases
a single freerunning sweep period by At. The subsequent cycles are
back to the normal time, but each is delayed by At sec. However, once
the sweep is synchronized by terminating it at a point on the sync
signal that has a slope of opposite sign to the sweep slope (in this case, on a
point of negative slope), then the time perturbation rapidly damps out.
Figure 5116 shows how the sweep returns to its original firing point
within several cycles of the original disturbance.
ti ti+M Zti 2t!+&t
Fig. 511. (a) Sweep perturbation — no sync signal; (b) sweep perturbation — sync
signal present.
The reason for the rapid damping can easily be seen when we expand
two adjacent cycles of the sweep for a closer look at the end point (Fig.
512). At the first firing point, the voltage jitter AEf produces an
increase in the sweep period of At. Thus the second cycle starts At later,
but because of the geometry at the intersection of the two straight lines,
this cycle ends only At' later. The time variation is reduced by a factor
S, at the instant of firing, where
& =
At
(523)
When the two intersecting curves have slopes of opposite sign as in
Fig. 512, 8 will always be less than unity. The third sweep starts
160
TIMING
[Chap. 5
At' later, and at the instant of firing, this is also reduced by the factor 8.
After n sweeps, the total time displacement from the normal time will be
5" At and
lim S" At =
n— ♦ large
Thus the sweep eventually returns to its original firing point with respect
to the sync signal. We see from Fig. 512 that the steeper the slope
of the synchronizing signal at the point of sweep intersection, the smaller
the value of 8 and the fewer the number of cycles required for sweep
recovery time. We further conclude that the optimum point to effect
synchronization (with any input signal) is where the slope is the steepest,
First cycle Second cycle
Fig. 512. Expanded sweep firing point showing stability.
such as the crossover point of a sine wave. If a choice presents itself,
the optimum sync signal would be a square wave or pulse train where the
sweep can terminate on a point of infinite slope.
Suppose that the sweep intersects the sync signal at a point that has
the same sign of slope; i.e.., both are positive (Fig. 513). Under these
circumstances any slight time perturbation At increases at the point of
firing by the factor 8' to At".
8'
= ^'>l
At
(524)
Instead of damping out, the change in sweep time now builds up geo
metrically as (8')" over n cycles. Eventually, as the sweep shifts its
relative position, it intersects the sync voltage at a point of negative
slope. If the sync signal has the proper time and amplitude relationship,
with respect to the sweep, synchronization will now be effected.
Sec. 56]
SIMPLE SWEEPS AND SYNCHRONIZATION
161
66. Regions of Synchronization. Synchronization is not automat
ically guaranteed upon the injection of just any external signal.
The sweep will not be pulled in unless the sync amplitude and period
happen to fall within regions welldefined in terms of the sweep constants;
regions, which we shall see below, are also a function of the sync signal
waveshape. As an initial example, refer to Fig. 514, where a sym
metrical square wave of variable amplitude is used as the control signal.
First cycle
Fio. 513. Unstable sweep synchronization
Second cycle
expanded firing region.
Fio. 514. Squarewave synchronization.
The first sweep fires when it terminates on the square wave at 3t„ and
if the input remains unchanged, the sweep will continue to fire once every
3 cycles. However, the increased square wave causes the second and
third sweeps to end at nonintegral multiples of the sync signal period, and
therefore they are unsynchronized. The fourth cycle is again synchro
nized, this time at the next smaller integral multiple of the sync period, at
2t.. We might also note that any freerunning sweep bounded by the
dashed lines shown in the first sweep cycle, that is, t a < t\ < h, intersects
162
TIMING
[Chap. 5
the changed firing voltage along the same portion of the square wave
and also fires at 3t,.
Let us now assume that the control signal is a square wave having a
constant amplitude of E, volts peak value. Then if its period falls
anywhere within the two extremes given in Fig. 515a and b, the fixed
freerunning sweep will fire over the same integral number of sync
cycles, over n cycles. At the one limit shown in Fig. 515a, the sweep
period becomes foreshortened because a portion of the sync signal
extends below the original firing line. Figure 515& illustrates the possible
lengthening of the sweep as a consequence of the effective increase in the
firing voltage.
E,+E,
E,h
E/—E s
—
*
—
—
D
B
— a
i
i
i
1
i
I
i
i
i
i
EJC,
h t
(a)
Fig. 515. (a) Minimum sync period producing synchronization; (b) maximum
sync period producing synchronization.
We shall first concern ourselves with expressing the lower usable limit
of the sync period (Fig. 515a). Since the two sweep triangles OBC and
ODE are similar, the ratios of their equivalent sides must be equal.
OE BE
OC BC
Hence one limit becomes
IT
nt,,.
Ej — E,
E,
K'l)
(525)
The solution of Fig. 5156, also by similar triangles, yields a second limit,
t„
k,m». _ 1 ( , . E\
h n\ 1 ^ E,)
(526)
If the sync amplitude is now increased, the sweep will stay in syn
chronization until the corner of the preceding half cycle intersects the
rising sweep, causing premature firing. This condition is indicated, for
both cases, by the dashed lines of Fig. 515. Consequently, for any
Sec. 56] simple sweeps and synchronization 163
value of nt„ the following condition on the maximum allowable sync
amplitude may be written
■ni,  \bU _ E,  E. M
or
U
E„,,
E,
= 1 
E f
2w  1 l„
2 h
(527)
Equations (525) to (527) are all straight lines in terms of the normal
ized coordinates t,/h and E,/E/. Each delineates one set of the boundary
1.4
1.3
1.2
l.l
1.0
0.9
0.8
«T 0.7
0.6
0.5
0.4
0.3
0.2
0.1
'« ma*
>i
^smax
V *
's min
n4
^
0.1 0.2 0,3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
I—
Fig. 616. Regions of synchronization— squarewave input signal.
values which limit possible synchronization. By plotting these equations
with n as a parameter, we define the regions of synchronization for a
squarewave input (Fig. 516). Any combination of periods and voltages
lying within a region ensures proper synchronization over n cycles of the
square wave. And to allow for possible fluctuation of sync and sweep
voltage amplitude, and sweep period, the best combination of parameters
lies at the center of any one region. Furthermore, the area of each
region decreases extremely rapidly as n increases, with the consequence
164
TIMING
[Chap. 5
that it becomes very difficult to ensure synchronization over ratios
greater than 4 or 5. The expected component variations would easily
shift the operating point out of the small synchronization regions holding
for large n.
All sync signals that effectively increase and decrease the firing voltage
by equal amounts, e.g., sine waves, triangles, sawtooths, etc., will have
two limits of their synchronization regions defined by Eqs. (525) and
(526). Equivalent limits for nonsymmetrical signals may be calculated
quite easily from their geometry.
For example, if the square wave used above is differentiated and
the train of equalamplitude positive and negative pulses is applied,
then only the negative pulse will
influence the timing. The sweep
cannot be lengthened since it will
always terminate on the negative
pulse or along the original firing
line, as in Fig. 517. In this case
Eq. (526) reduces to
t.„
1
Equation (525) will still hold as
a second limit. A new equation
must be found to delineate the third
boundary (Prob. 514).
Equations (525) and (526) may
also be applied to nonsymmetrical
waves by properly interpreting the various voltage terms they contain.
We rewrite these equations
Fig. 517. Synchronization with pulses.
t«,min _ I 1 ZZJ 1 — I
h ~ n\ l Ef)
tg.max _ f 1 _J_ 8p \
h n\ y+ E,)
(528)
(529)
where E,„ is the negative peak of the sync signal and E, p is the effective
positive peak, both measured from the original firing line. Of course, as
with the pulse, the sweep must be able to terminate at these voltage
values.
The difficult problem in finding the regions of synchronization is the
evaluation of the third boundary.
E,, m „ _ * /tA
Sec. 56]
SIMPLE SWEEPS AND SYNCHRONIZATION
165
For signals having simple geometric waveshapes, it is possible to find an
explicit analytic solution, but for more complex signals we would have
to turn to a tedious graphical construction.
We can appreciate the increase in complexity by treating the case of
triangularwave synchronization (Fig. 510). Two limits are known
[Eqs. (525) and (526)], and the third will be found from Fig. 518, which
illustrates the conditions existing at the maximum possible value of the
sync signal. If we assume that the sweep is still properly synchronized
but on the verge of limiting,
nt,
h
E, + KE., L
E,
(530)
where K is a positive or negative constant relating the point of inter
section to the peak value of the triangular wave. The sweep starts
Fig. 518. Triangularwave synchronization — maximum value of sync signal.
and ends when it intersects the sync wave M a sec before the triangular
wave crosses the quiescent firing line. From the similar triangles ABC
and DEC,
KX.
4
M a
(531)
When the sync voltage becomes infinitesimally larger than the value
assumed for Eq. (530), the sweep no longer clears the negative peak.
It now fires prematurely upon intersecting the sync signal 3<„/4 — M a
sec before the original firing point. Thus the following ratio now holds :
nt.  (3J./4  M a ) _Ej  E,, m
E,
(532)
Substitution of Eq. (531) into (532) provides us with a second equation
relating t,/h and J5, imml /JS/, one which, however, also includes the third
variable K. The simultaneous solution of Eqs. (530) and (532),
166 timing [Chap. 5
eliminating K, completes the solution, with the final answer given in
Eq. (533).
ny*  y + (4n  3)xy  4x + 4z 2 = (533)
where
Equation (533), the equation of a conic section, defines the third
boundary of the synchronization regions which are plotted in Fig. 519.
1.4
1.3
1.2
1.1
1.0
* 0.9
0.8
'. 0.7
0.6
0.5
0.4
0.3
0.2
0.1
/
'smax
/
u
/
71"
■ 1
E
(,
S
^
*.
s
M
l
V
M
V
•^
1—
*
J
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fig. 519. Synchronization regions — triangularwave input signal.
The larger areas indicate that synchronization will be maintained in
face of widerlatitude parameter variations than can be tolerated for
the square wave (compare Figs. 516 and 519). However, a square
wave or pulse is vastly preferable because of the fast recovery from sweep
perturbations.
The two waveshapes above were presented primarily in an effort
to establish graphically the interlocking conditions which must be evalu
ated and satisfied for precise and accurate output control. We made
no pretense of covering the multitudinous possible sync signals, such as
Sec. 56] simple sweeps and synchronization 167
pulses, sine waves, etc. When order of magnitude or approximate
boundaries will serve our purposes, the replacement, for calculation, of
complex waveshapes by simpler ones suffices. For example, a sinusoidal
signal might be approximated by a truncated triangular wave, leading to
regions slightly smaller than those of the triangle and slightly larger
than those of the square wave. If we are interested in only one param
eter's variation, then evaluation from a sketch gives the allowable limits
directly.
Synchronization is employed in many circuits other than the thyratron
sweep. In complex systems, often all waveform generators are controlled
by one master, which is made extremely stable with respect to time
changes. The process of synchronization imparts this same stability
to the remaining circuits. Each freerunning generator, oscillator, multi
vibrator, phantastron, etc., has its own mode of synchronization con
tingent on its individual control characteristics. Consequently, the
individual regions will be well defined but will differ from one circuit to the
other. When needed, they may be calculated by application of the meth
ods discussed in this chapter, modified as necessary to suit the particular
problem.
Example 61. A signal having a highly complex structure with many jagged peaks
must be used to synchronize a sweep over a ratio of 2:1. In its original form, the
input is not suitable for synchronization, but by amplifying and clipping, it can be
converted into a square wave. And, if needed, we can differentiate it to obtain
pulses or integrate it to obtain a triangular wave. The questions of interest are:
(a) Which signal is best for synchronization?
(6) At what point should synchronization be effected?
Solution. For optimum operation the sweep should be synchronized so that the
nominal operating point is in the center of the appropriate region. By this we mean
that it should be located as far as possible from the boundaries, allowing the time and
voltage to vary over the widest possible limits before the sweep becomes erratic. In
general, the operating point which permits the greatest variation in the time ratio will
not be the optimum one with respect to voltage changes. However, from Figs. 516
and 519 we see that the time bounds are much narrower, and therefore we shall
usually design the circuit for the best response with respect to time variations.
1. For squarewave synchronization, the plot of Fig. 516 indicates that the widest
changes in t,/t\ occur along a line through the apex of the region n = 2. The bounds
are
0.425 <^< 0.575
and the nominal freerunning sweep should be set at the center, i.e., at t, =» Q.5U.
Assuming a constant sweep time, the sync signal can increase from 0.15E/ to 0.252?/
or can decrease to zero before the sweep becomes unstable.
2. With the triangularwave input, the maximum permissible change in U does not
occur along a line through the apex. In Fig. 519 we see that at E./Ef = 0.3,
0.35 <r< 0.53
168
TIMING
[Chap. 5
The freerunning sweep should be adjusted to the center value h — 2.27*,. This
establishes the following nominal limits on the sync amplitude from the operating
point given above:
0.12J?/ < E, < 0.5JJ,
3. The regions which might be drawn for pulse synchronization (Prob. 514) would
indicate that the optimum operating point should lie at
t.
0.375
E, = 0.5jB/
Permissible time and voltage variations are between the limits
0.25 < r < 0.5
0.25 < ^ < 0.625
To summarize, the percentage variations of freerunning sweep time and sync voltage,
as related to the nominal value of U and Ef, are:
Percentage variation
Square
Triangle
Pulse
At
U
ae,
E f
±15
15, +10
±20
18, +20
±33
25, +12.5
We conclude that pulses are best for synchronization purposes. Besides allowing the
widest tolerances in time, their steep edges force recovery of any perturbation within
one, or at most two, cycles. The triangular wave does permit wider latitude to the
sync amplitude, but the difficulties involved in its generation as well as the longer
recovery time would usually preclude its use; pulses or even the square wave is to
be preferred.
PROBLEMS
61. We have at our disposal a coldcathode gas tube with the following character
istics. The tube fires at 90 volts and maintains a constant drop of 50 volts while
ionized. It requires at least 50 ixa to sustain conduction and would be damaged by a
current in excess of 200 ma. Using a power supply of 200 volts, compute the mini
mum values of Ri and R t that may be employed. Adding a safety factor of 20 per
cent to the resistor values found, compute the value of C necessary to give a 100cps
sweep.
62. The circuit of Fig. 520 is a schematic representation of a basic current sweep.
The switch stays in position 1, while the coil current rises from its initial value toward
I„. At the firingcurrent value I/, the switch is thrown to position 2, discharging the
coil. Once the current falls to its initial value, the switch is returned to position 1
and the cycle repeats.
(a) Draw the current waveshape il, indicating all time constants.
(6) If /„ = 200 ma, /< = 80 ma, and Gi = 0.001 mho, what values must be chosen
for // and Ri if we wish to produce a triangular waveshape?
SIMPLE SWEEPS AND SYNCHRONIZATION 169
(c) When the inductance is 1 henry, to what must Gi and Ri be changed if the tri
angular sweep period is to be 2 msec?
Fig. 520
53. We have a thyratron with the following characteristics: /max  1 amp, lbx =
100 )>&, Ef = 10 — 5e e , and E a = 10 volts. This tube is used in a circuit similar to
Fig. 55, where E a = 300 volts, Ri = 3 megohms, E e = 20 volts, C = 0.001 /if.
(a) What is the sweep frequency of this circuit?
(6) For the optimum sweep response, what value should be assigned to i2 s ?
(c) With R 2 as given in part 6, what is the sweep recovery time?
54. A 884 thyratron has a firing characteristic which may be approximated by
Ef = 16 — 10e e . Its maintaining voltage E a is 16 volts, and lbx is 200 pa,. Design a
freerunning sweep of 100 pseo duration that will be not more than 5 per cent non
linear. Use a power supply of 250 volts and the minimum value of sweep resistance.
In the interests of simplicity, assume that the discharge time is insignificant. Specify
all circuit values and all sweep voltages.
55. We wish to use the thyratron of Prob. 54 in a circuit that will generate an
approximately triangular waveshape. (The rise and fall times are identical.) The
freerunning peaktopeak output amplitude should be 50 volts.
(a) If the available power supply is 300 volts, and if J2i = R t , calculate the initial
and the firing voltage for this sweep.
(6) Express the period as a function of the time constants and voltage values.
66. Design a driven sweep which makes use of the thyratron of Prob. 53. The
sweep is to have a duration of 300 /isec and an amplitude of 50 volts and is to be trig
gered by a positive pulse 1 volt greater than the minimum amplitude. Use a power
supply of 300 volts and a capacitor of 10  ' farad in your sweep.
(o) Find suitable values for E c , R u R t , R 3 , and R t in Fig. 57. Express Ri and Rt
as a ratio rather than as actual resistance values.
(6) Calculate the minimum pulse amplitude for triggering.
(c) How long after the pulse is applied will the sweep start?
57. The circuit shown in Fig. 521 is a triggered sweep operated by the 20volt
positive input triggers. It uses an 884 thyratron whose characteristics are given in
Prob. 54.
(o) What is the minimum spacing of the trigger pulses U so that each trigger will
initiate one complete sweep?
(6) How long after the trigger is applied will the sweep start?
(c) Sketch the output waveshape, giving all values and times.
68. Assume that the input impedance appearing from grid to cathode of the sweep
circuit shown in Fig. 521 is 50 n/t! in parallel with 1 megohm before firing. What is
the narrowest pulse that would institute a sweep? Sketch the voltage appearing
from grid to cathode if, after firing, the gridtocathode resistance falls to 2,000 ohms.
69. (a) Calculate the output sweep period and amount of nonlinearity for the
circuit of Fig. 522. The initial value of the sweep is 20 volts, and the thyratron fires
when the capacitor voltage reaches 100 volts,
170
TIMING
[Chap. 5
20v
»
to
' *
At
*
*!—
t
5M<
^VW*AAAr
,,20K 40K i+250v
1°
1
(6) Replace the triode by a resistor that will give the same sweep time and ampli
tude. Recalculate the linearity and compare with the results obtained in part a.
Fig. 522
510. A simple thyratron sweep starts charging from E { = 20 volts toward 250
volts, and the tube fires at E, = 100 volts.
(a) Calculate the sweep NL.
(6) Considering the following parameters, one at a time, find the percentage change
in m, E lt Ef, E c , and £» which will result in a 1 per cent increase in sweep nonlinearity .
(Hint: In each case expand the natural logarithm into a series.)
(c) List these parameters in the order that they affect the sweep quality. Assign
the value of 1 to the term having the most pronounced effect and give proportional
values to each of the other terms.
611. (a) Repeat Prob. 510 with respect to the sweep duration; i.e., calculate the
permissible changes in the various parameters, taken one at a time, which will cause
a 1 per cent change in the sweep duration.
(6) By reference to other texts on gas tubes, discuss the effect of temperature
variation on the sweep duration. Is there any way to minimize this problem within
the expected range of 30°C change in ambient temperature?
512. Derive an expression for the percentage nonlinearity of the sweep, starting
from the straightline representation of Fig. 58a. Express your answer in terms of
the sweep period and the time constant and also in terms of the critical voltages.
513. Verify Eq. (522).
514 Plot the regions of synchronization for a train of equally spaced positive and
negative pulses. Assume that the pulses are extremely narrow. Explain your
answer.
516. A 3volt peaktopeak square wave is injected at the grid of a thyratron sweep.
The tube is biased to fire at a nominal value of 100 volts, and the drop across the con
ducting tube is constant at 25 volts. We can express the firing characteristic as
E f  20  10e„.
(a) If the freerunning sweep period is 500 /jsec, what range of sync frequencies will
cause the sweep to lock in over a ratio of 2 : 1 ? How does the amplitude of the sweep
extremes compare with the nominal freerunning values?
SIMPLE SWEEPS AND SYNCHRONIZATION
171
(b) When the squarewave period is 400 jisec, what range of the syno voltage will
effect synchronization? Calculate the bounds by means of a graphical construction
and then check the results against Fig. 516.
516. A perfectly linear sweep has a normal period of 1.0 msec. This can be
expected to vary by as much as ± 10 per cent because of the variations in the firing
voltage. In the absence of synchronization, the sweep starts at zero volts and termi
nates at the nominal value of Ef — 100 volts. The retrace time is zero. It is
required that the sweep be synchronized so that its period is exactly 1.0 msec despite
the instability in amplitude.
Three synchronization waveshapes are shown in Fig. 523 plotted at the firing line
of sweep. In each, (, is 1.0 msec or some integral submultiple.
(a) Which among these will provide the desired synchronization?
(6) When the sweep is synchronized under the conditions specified, what is the
possible number of cycles of the synchronizing wave over which synchronization can
take place?
e.i
««2
e,3
«. H i
Ti
T
h>~* *!*■"* \ 15v
15 v
W
1 i
15 v
45 v
Fia. 523
617. (a) Verify Eq. (533) by direct solution from the conditions of triangular
wave synchronization.
(6) Calculate the slope of Eq. (533) at x = 1 as a function of n and compare it
with the slope of the t,, m in/ti line of the n + 1 region.
618. Consider that the signal available to synchronize a sweep at 10 kc must be
derived from a 50kc sinusoidal oscillator. It will be clipped to a symmetrical square
wave and may also be differentiated. Calculate the optimum freerunning sweep
period and the optimum ratio of sweep to sync signal. Justify your answer. What is
the maximum number of cycles over which the sweep can be synchronized if the free
running sweep has ±3 per cent jitter?
BIBLIOGRAPHY
MacLean, W. R.: The Synchronization of Oscilloscope Sweep Circuits, Communicor
tions, March, 1943.
Millman, J., and H. Taub: "Pulse and Digital Circuits," McGrawHill Book Com
pany, Inc., New York, 1956.
Puckle, O. S.: "Time Bases," 2d ed., John Wiley & Sons, Inc., New York, 1951.
CHAPTER 6
VACUUMTUBE VOLTAGE SWEEPS
61. Introduction. Even a perfunctory reexamination of the simple
gastube sweeps of Chap. 5 testifies to their inherent unsuitability for
other than the most nonoritical applications. Generation of veryshort
duration sawtooths, smaller than about 20 usee, was precluded by the
circuit's excessively long recovery time. Instability, introduced by the
dependence of the ionization potential on external factors, temperature,
noise, etc., limited the longest sweep duration to approximately 0.1 sec.
And only through severely restricting the amplitude could sweep linearity
be maintained within acceptable tolerances. Consequently, when strin
gent specifications must be met, we are forced to look for other, better
techniques of sweep generation.
In this chapter each of the three sweep essentials, rise, recovery, and
switching, will be separately treated with a view toward optimizing the
overall performance. The first sections are concerned with the establish
ment of an almost ideal, linear charging path. For this purpose, it is
necessary to employ active circuit elements, which we shall, in this
chapter, limit to vacuum tubes. Three methods of linearization, all
involving feedback, are in common use: current feedback for constant
current charging of the capacitor, positivevoltage feedback wherein the
needed correction term is first developed and then applied, and finally
the effective multiplication of the time constant and charging voltage,
through the use of negativevoltage feedback. After examining each one
of these methods of linearization, the improvement of the sweep recovery
time and the nature of the switching process will be considered. Later
sections will combine the individual circuits into two practical voltage
sweeps, the Miller integrator and the phantastron.
62. Linearity Improvement through Current Feedback. If we were
to examine the differential equation defining capacitor charging, we
would conclude that its terminal voltage rises linearly with time only
while being charged from a constantcurrent source. One method of
approaching this ideal situation is through the application of current
feedback, as illustrated in the sweep circuit of Fig. 6la. Here the
complete capacitor current flows in R K , producing a proportional volt
172
Sec. 62] vacuumtube voltage sweeps 173
age drop, which, together with E, determines the gridtocathode voltage
(e„t = E — i b RK) Any change in current automatically establishes the
conditions which act to return the current flow to its original value.
Quite possibly some factor may cause a momentary decrease in i b , and the
reduced current will, of course, reduce the voltage drop across Rk But
this increases e„*, which, by inducing a resultant increase in tube current,
effectively opposes the original change. An increase in load current
produces just the opposite effect, again with the tube aiding in the effort
to maintain the charging current constant.
This circuit, from grid to cathode, is simply a cathode follower. There
fore, provided that the plate voltage remains sufficiently high (maximum
capacitor voltage limited), the grid will not draw current and we can
assume unity gain as a reasonable first approximation. The drop across
/<*«*
(a) (b)
Fig. 61. (o) Sweep using current feedback for linearization; (6) equivalent circuit.
R K is identically the gridtoground voltage E and it logically follows
that the charging current must be
*& =
Rk
50
70 K
= 714 n&
(61)
In order to verify this rough approximation, we might calculate the
TheVenin equivalent circuit across C. Referring to Fig. 616,
R™ = r p + (m + 1)Rk = 70 K + 7,070 K (62a)
Etk = En + nE = 300 + 5,000 volts (626)
With a highju tube, nE » #», (m + 1)R K » r p , and Eq. (61) gives a
reasonably correct answer. The actual shortcircuit current Eth/Rtk
is only 4 per cent larger. Constraints, which are considered below,
limit the maximum output sweep voltage to a very small percentage
of the total Thevenin equivalent value given in Eq. (62). In spite
of the actual exponential charging toward E T h with the long time con
174 timing [Chap. 6
stant (n + 1)RkC, the current varies but slightly over the restricted time
range of interest.
The plate voltage falls from En, at a rate determined by the capacitor
charging. Substitution of the approximately constantcurrent equation
(61) into the charge equation yields
1 /" F
e 2 = En  ^ / % dt = E»  R —^ t (63)
Equation (63) remains valid until the grid begins conducting. Its load
ing of the constantvoltage source will change the charging current and
adversely affect the sweep linearity. The bottoming value of plate
voltage is found from the circuit of Fig. 616 by setting e gk = 0. At this
point, the actual voltage drop across Rk is E, and since the current that
flows through Rk also flows through r P , the lower bounding value of the
Linear sweep output becomes
( I + fc)
Eu = V 1 + R J ® = 10 ° volts (6 " 4)
The capacitor discharge mechanism is usually triggered when the out
put falls to some preset voltage. In a fixed duration sweep, this might
be the bottoming value found in Eq. (64). If, however, the charging
current is adjusted by varying R K , then the saturation voltage will also
change. Decreasing R K increases the current, thus increasing the charg
ing rate and shortening the sweep. But any reduction in R K also
affects the possible sweep amplitude by raising the plate bottoming
voltage. The discharge point must correspond to the worst condition,
i.e., the solution of Eq. (64) for the smallest R K . Hence, from Eq. (63),
the sweep time becomes
it RkC E »~ E ' u (65)
where, for example, when Rk.™* = 35 K, E' u = 150 volts.
As the plate falls from 300 to its bottoming value of 150 volts, the
capacitor charges by only 150 out of its equivalent steadystate value of
5,300 volts. Hence the sweep nonlinearity may be found from Eq.
(522).
1 "SO
NL  POT) 125 %  ° 36 %
In an equivalent RC sweep, where C charges by the same amount from
a constant 300volt source, the nonlinearity [Eq. (521)] is
NL=125%ln 300^150 = 8  7 %
The improvement is quite impressive.
Sec. 63]
VACUUMTUBE VOLTAGE SWEEPS
175
63. Bootstrap Sweep. Bootstrapping, a form of positive feedback,
linearizes the sweep by first determining what voltage correction term
is needed and then deriving it through acting on an already existing
voltage. The necessary circuit modification usually follows directly
from the analysis of the original problem posed. For example, the non
linear charging in the RC circuit of p
Fig. 62 is a consequence of the
dependency of the loop current upon
the capacitor voltage.
E'  e e
R
(66)
Fig. 62. Simple RC sweep.
If, somehow, we could cancel the
effects of e c , through the adjustment
or replacement of one of the circuit components under our control, then
the desired linear charging would be assured. One obvious course is to
replace E' by E + e c . This required drive can be developed quite
easily by amplifying the capacitor voltage and connecting the charging
resistor to the amplifier output instead of to the battery. Because of the
special charging requirements, the output must have a dc level of E.
Thus it follows that Fig. 63 represents the general form of the bootstrap
sweep. In this case positivevoltage feedback is employed to increase
the effective charging source voltage at the same rate as the increase
in the sweep voltage e c .
ft
l_
C± + *
R
VW
>E
E+Ae,,
A<1
"0 «o *1 «a' *
Fio. 63. General bootstrap sweep and output waveshapes.
Constantcurrent charging of C is contingent on the amplifier gain A
remaining unity. Any variation is immediately reflected as a change in
the output waveshape and sweep period (Fig. 63). This dependency on
A may, if serious, prevent the use of the bootstrap circuit for all but non
critical applications.
The sensitivity of sweep duration with respect to the voltage ampli
fication may be interpreted quantitatively by starting from the equivalent
circuit controlling the charging current flow. Referring to Fig. 63 and
176 TIMING
writing the input node equation yields
E + Ae c — e c
R
E
R
(1  A)e c
R
[Chap. 6
(67)
The righthand side of Eq. (67) represents the circuit as seen by the
capacitor: it consists of a constantcurrent generator in parallel with its
own internal admittance (Fig. 64).
We might observe, from Fig. 64, that when A < 1, the charging of C
is exponential rather than linear. The final steadystate voltage, deter
mined by setting i c = 0, is E/(l — A), and by inspection of the circuit,
we see that the charging time constant is CR/(l — A). Therefore the
sweep voltage may be expressed as
«.(0 =
E
1  A
(1
RC 1 )
(68)
However, when A = 1, G Q becomes zero and C is charged directly from
the ideal constantcurrent source.
A = 1
_E_.
e °~ RC 1
(69)
Suppose that a change in either tube or circuit parameters makes A
slightly greater than unity. Under these circumstances G„ becomes
negative and the terminal voltage
increases exponentially toward in
finity. The equation defining this
region of operation is the same as
Eq. (68). But A > 1 means that
the time constant is now negative
and that the system's single pole
will move along the real axis into
the right half plane. Since a physi
cal device cannot handle infiniteamplitude signals, any slight disturbance
immediately drives the amplifier toward saturation.
In order to see how the sweep time depends on A, we first calculate
the sweep duration from Eq. (68).
Fig. 64. Bootstrap equivalent circuit.
t' =
RC
1
In
E
E  (1  A)E,
No generality is lost by choosing a convenient termination point, one
which will keep A in the forefront and at the same time eliminate super
fluous voltage terms. The greatest simplification occurs by setting
Sec. 63]
E, = E:
VACUUMTUBE VOLTAGE SWEEPS
ta
RC . 1
r=i ln A
177
(610)
As a basis for comparison, the duration of the linear sweep of the same
amplitude is evaluated from Eq. (69).
h = RC
And from Eqs. (610) and (611),
ta = 1
h 1 
A A
(611)
(612)
For A = 0.95, 5 per cent below the optimum value of unity,
j = 1.082
or the sweep time is now 8 per cent longer than the perfectly linear sweep.
When A — 1.05, the sweep duration is reduced to 0.976<i. The separa
tion from the linear sweep with respect to changes in A becomes even
Fig. 65. Practical bootstrap amplifier.
more pronounced as the sweep amplitude increases. From the above
discussion, the only possible conclusion drawn is that the gain of the boot
strap amplifier must be highly stabilized if we are to avoid sweepduration
instability. But even with the high degree of sensitivity of the sweep
period to changes in A, the sweep linearity and maximum sweep ampli
tude are still superior to what it is possible to achieve with a simple RC
network.
A practical, directcoupled amplifier for bootstrapping applications
appears in Fig. 65. Negativevoltage feedback, applied through the
resistor network R a and Rf, both sets and stabilizes the overall positive
178
TIMING
[Chap. 6
gain. This amplifier's gain varies by less than 0.01 per cent, with regard
to expected tube and parameter chaiiges from its nominal unity gain
value. Bz controls the output voltage level, and thus the charging
current: it might also serve as the fine sweep control. An obvious
circuit disadvantage is the need for two power supplies which, for optimum
gain stability, must be highly regulated.
64. Miller Sweep. The Miller sweep, which utilizes negativevoltage
feedback for sweep linearization, poses no problems as to poles in the
right half plane. And, in addition, it generates a sweep virtually inde
pendent of the amplifier gain. Consider the basic circuit of Fig. 66a,
(a)
Fig. 66. (a) Basic Miller sweep; (fc) Miller sweep — equivalentcircuit representation.
where the amplifier has a forward voltage gain A and an output imped
ance R,. If, as was done for the bootstrap, we write the current equation
at the input node,
. _ e, — Ae,j
%i ~ R, + 1/pC
We now, by solving this equation, find the input impedance,
„ e ( _ R.
+
i
pC(l  A)
(613)
Equation (613), the Miller effect, expresses the effective multiplication
of the feedback capacity and reduction of resistance due to the negative
voltage feedback. With current feedback (the circuit of Fig. 61), the
opposite effect appeared. It follows, from Eq. (613), that the input
circuit may be represented as shown in Fig. 665. If the current flow
from the output back to the input is kept small, then the drop across R,
Sec. 64] vacuumtube voltage sweeps 179
due to the feedback component may be neglected. R controls the charg
ing rate as well as limiting the feedbackpath current flow; it is usually
very much larger than R,. We have thus effectively isolated the input
and output circuits, allowing an independent solution for the input
voltage. It may then be reflected into the output by multiplying by
the circuit gain e = Ae t .
Since we are interested in generating a linear sweep, charging must
be limited to a small portion of the total exponential. The input circuit
will be treated as an integrator with the reflected resistance R,/(l — A)
contributing an additional small constant term.
R, 1 /"'
e% = J5 — : — ttA jt E H — 7 r — / E dt
R  + m  A) (« + r #i)«'*>i
And the amplified input voltage appears at the output:
. AR.E AE
6 ° = Aei = B. + g(l  A) + ' — ' (6 ' 14)
(« + A)
C(l  A)
If the amplifier gain is a very large negative number, then (1 — A) ^  A\
and A cancels out in the sweep term of Eq. (614). This necessary
condition must be satisfied in order to make the sweep relatively inde
pendent of the amplifier parameters. As a further result of the large
gain, R » R,/(l  A) and
_ R, „ E
e 'R E RC t
If the much more stringent requirement that R ^> R, is also satisfied,
then this equation may be reduced even further:
eo^^t (615)
Equation (615) has the same form as the charging equation derived
for the bootstrap [Eq. (69)]. But while the linear charging in that
circuit was critically dependent on the value of the amplifier gain, this
is not true in the Miller sweep, where the only requirement is a large
negative gain.
An amplifier with a gain of —100 and source voltage of 300 volts
generates a sweep of 100 volts, while the input only changes by 1/A times
as much, or by 1 volt. Since this is a very small portion of the possible
300volt charging curve, the sweep will be extremely linear; nonlinearity
180 timing [Chap. 6
of better than 0.1 per cent is easily achieved, and with a veryhighgain
amplifier even 0.01 per cent nonlinearity becomes possible.
Triode Miller Sweep. The procedure used in calculating the circuit
waveshapes and in subsequent verification of sweep linearity is outlined
below for the triode Miller sweep of Fig. 67o. The solution follows
in time sequence the behavior of the circuit. In each region we draw the
model holding and use it to evaluate time constants and boundary
conditions.
Along with the general solution outlined below, we shall consider
the specific components of Fig. 67 to illustrate the order of magnitudes
involved.
g P, p R ' ,,r 
±
rntrr
T
(a) (b)
Fig. 67. (a) Triode Miller sweep; (6) model holding in the active region. For the
purposes of illustration, the following typical values are assigned to this circuit : in = 60,
r„ = 10 K, r c = 400 ohms, Rl = 50 K, R = 1 megohm, C = 0.001 M f, and En = 250
volts.
Before closing the switch at t = 0, the grid, returned through R to En,,
is conducting. Since the 400ohm grid resistance is much less than R,
the capacitor charges to Eu, with the polarity indicated. The grid is not
actually at zero but at some small positive value, usually a fraction of a
volt, as given by Eq. (616).
400.
E c (0 ) = 5 E» —
10 6
250 = 0.1 volt
(616)
Once the switch closes, the first question of interest is, What is the
new value of grid and plate (point b) voltage? Our qualitative argument
as to the circuit behavior follows. The voltage across the capacitor
cannot change instantaneously; therefore any change at the plate must
also appear as an equal change in grid voltage. When the switch closes,
the grid either rises, falls, or remains constant at its slightly positive
value. We shall first assume that it remains unchanged. The plate
current corresponding to the positive grid voltage is very large,
causing a large voltage drop at point b which, coupled by the capacitor,
Sec. 64]
VACTJTJMTUBE VOLTAGE SWEEPS
181
appears at the grid and cuts the tube off. Hence a contradiction arises,
and the original assumption must have been incorrect.
This contradiction may be resolved by observing that only a small
change of plate voltage may be tolerated without cutting off the tube.
The plate current flow, after switching, must be small, and we conclude
that the tube is driven almost to, but not quite into, cutoff.
The operation in the active region, including the initial drop at both
the plate and grid, may be calculated from the model which holds
immediately after switching. In Fig. 676, the complete circuit, seen
looking into the plate of the tube, was replaced by its Thevenin equiv
alent. E h , is the saturation value of the plate voltage (e„* = 0), and A
=£n —I—
=T=C(1A)
1 !
I A
(a)
(b)
Fia. 68. Triode Millersweep models, (a) Grid circuit; (b) plate circuit.
is the voltage gain in the active region. From the unloaded platecircuit
model,
mKl „ _ r p w (617)
A =
r P + R L
Eh,
r p + Rr,
Eh
With the parameters given, the equivalent platecircuit resistance
(R, = r p  R L ) is only 8.3 K, the gain is —50, and E b , = 41.7 volts.
Following the argument presented with respect to the general Miller
sweep, the complete timing can be reflected into the grid circuit (Fig.
68o). The only question remaining unanswered is, What is the initial
voltage across the reflected capacity C(l — A)? Just after switching,
the loop voltage from grid to ground can be found from Fig. 676.
#0(0+) = #„, + I c (0+)R, + AE C {Q+) + E„,
Solving for the grid voltage yields
E c (0+)
— Ew, \ Eb,
1  A
+
R.
1
/c(0+)
(618)
The first term in Eq. (618) is the voltage across C(l  A) immediately
after the switch is closed.
Since the current through R remains invariant across the change in
models from Fig. 676 to Fig. 68a, the second term of Eq. (618) can be
182 timing [Chap. 6
identified as the drop across the reflected source resistance. The current
which charges C(l — .A) is
E»  B c (0+)
/c(0+)
R
but E c (0 + ) must he within the small grid base during the entire sweep.
Hence En, ;» E c and
Ic £* ^ (619)
The almost constant charging current indicated by Eq. (619) proves
that the sweep produced will be highly linear.
By substituting the terms from Eqs. (617) and (619) into Eq. (618),
we can also express the initial grid voltage as
^ 0+ > =  r, + Of; 1)B L *» + 1^2 f <**»
In this example, the reflected source resistance is only 160 ohms,
which is very much less than the 1megohm timing resistance. The
second term of Eq. (620) is almost completely negligible; it is only
0.04 volt. With a high/* tube the significant first term of Eq. (620)
may be approximated as
E c (0+) = — J?£ = 4.1 volts (621)
M + 1
We might note that this is only slightly above the grid cutoff voltage of
— Etb/n ( — 4.2 volts). This verifies our previous contention that the
tube is driven to the verge of cutoff.
After the initial drop, the grid starts charging from E c (0 + ) toward
Em, with the long time constant
n = C(l  A)(r + j^t) = RC(l  A) (622)
With the constant charging current assumed in Eq. (619), the small
portion of the exponential rise used may be approximated by
«.(«) = E c (0+) + RC{ f *_ A) t (623)
The grid rises at a rate of Ej»/i?C(l — A) volts/sec from its initial value.
The abrupt change in voltage at the grid, as the tube turns on, is
A# cl = E c (0+)  E c (0~)
Since the plate and grid are coupled by C,
E t (0+) = Ea,  A£ c i = 246 volts
Sec. 64] vacuumtube voltage sweeps 183
The plate waveshape will simply be an amplified version of the signal
appearing at the grid. It follows that the plate will fall A times as fast
as the grid rises.
and since A is a large negative number, the plate rundown is independent
of the gain.
*(*) =&(()+) g* (624)
The next change in the circuit state occurs when the grid reaches
zero and starts conducting. From Eq. (623), the linear grid rise of
E e (0+) volts takes
h= ~ E ° { ° +) ti sec (625)
EtVb
With the values given in Fig. 67, the sweep lasts for approximately
800 jxsec. Only a small fraction of the 50msec grid time constant is
used for the linear plate rundown, and over this interval the sweep
nonlinearity is 0.16 per cent.
The plate voltage corresponding to zero grid voltage Eta may be
found in several ways. For example, the sweep time found from Eq.
(625) can be substituted into the platevoltage equation. A somewhat
more accurate method is to solve the platecircuit model of Fig. 686 with
e c = 0.
Ebo = IcRi + Et, = p Ebb + Eb, (626)
In the case where R, <3C R, the bottoming voltage is almost exactly E it .
In the example cited, Ebo = 43.7 volts, only 2 volts above Eb,.
The several answers for Ebo will not be in exact agreement because
of the approximations made in the course of the analysis. The value
found from the model is more nearly correct since it comes from^the basic
circuit instead of being the product of many intervening steps.
After the end of the linear rundown, the grid's conduction does not
remove the tube from its active region; it only reintroduces r c from grid
to ground in Figs. 676 and 68a, changing the time constant and the
steadystate voltage. From the Thevenin equivalent of the grid circuit,
the new values are
E„2 = •=; Ebb = E C (Q~)
K (627)
T2  C(l  A)
(+'fm
184
TIMING
[Chap. 6
In this region the 160ohm contribution due to the reflected source
impedance becomes significant compared with the 400ohm grid resist
ance. The new time constant of 28 /*sec is obviously much smaller than
n, and the rise rate is much faster. The positive grid resistance is
introduced into the circuit at zero volts at a point where it cannot
abruptly change the voltage anywhere within the timing capacitor loop.
Consequently the sweep will enter
the new region smoothly, without
any jumps appearing in either the
grid or plate waveshapes.
As the grid continues rising, the
plate falls proportionately, finally
bottoming at Et, mill . We evaluate
this voltage from the platecircuit
model (Fig. 686) by setting e c = 0.1
volt. Even in the positive grid
region, the ratio of the change in
plate voltage to the change in grid voltage is the circuit's amplification.
Thus E b , min might also be found by writing
E b , min = E b0 + AE e (0~) = 38.7 volts
Once the tube bottoms, the sweep is absolutely stable and remains
in this state until external conditions force a change. When we open the
switch, we remove the active element and must again draw a new model,
this time to define the behavior during recovery. Figure 69 illustrates
the conditions prevailing.
Just before, and therefore just after, switching, the voltage across C
is .Et,„i». The net change in the loop voltage will appear across the total
series resistance. It will divide proportionately across Rl and r c , but
remember that the jump at both the grid and plate must be equal. The
grid voltage jumps by &iE c , from E C (Q) to a new value, £„,„«.
Fig. 69. Circuit model holding during
recovery.
E c .
= Ec(O) +
r c + Rl
[E»
E,
b.min
E,(0)]
Rl
(Ett  £».„,„) = 1.69 volts
(628)
Point b rises by the same amount. Recovery now proceeds with the time
constant
t 8 = (Rl + r c )C S RlC (629)
toward the original steadystate voltages, E» and E c (0~).
The complete waveshapes, with all voltages, times, and time constants
indicated, are presented in Fig. 610.
Sec. 64]
VACUUMTUBE VOLTAGE SWEEPS
185
Complete recovery requires 4r 3 sec, or 200 usee, which is on the same
order of magnitude as the plate's linear rundown. Unless we do some
thing to shorten this time (Sec. 65), an unreasonably long interval must
elapse between sweeps.
Fig. 610. Triode Millersweep plate and grid waveshapes (not to scale).
Voltage Control over Sweep Time. External voltage control of the sweep
duration may be incorporated by coupling point 6 (the plate) of the
Miller sweep through a plate catching diode to the control voltage
E (Fig. 611). With the circuit in the normal, unexcited state, the
switch is open and the diode con
ducts, maintaining point b at E volts.
The grid voltage just after switch
ing can be found by following the
procedure previously discussed, with,
however, an initial voltage of E
across C. If R » R L \\ r P and the
contribution due the reflected resist
ance can be ignored, we obtain
E c (0+)
TpE,,,,  (r p + R L )E
(630)
Fig. 611. Circuit for voltage control of
sweep period — Miller sweep.
r p + ( M + 1)R L
Equation (630) expresses the linear
relationship existing between E c (0+) and E. Since the sweep time is
determined by the amount of the initial grid drop, it follows that the period
is a linear function of E. However, E must be restricted to the range
from Evb to Eh.. When E reaches its maximum value Em, Eq. (630)
186
TIMING
(Chap. 6
reduces to Eq. (621), and when E is set equal to E b „ E c (0 + ) becomes, as
expected, zero and the sweep time is also zero.
The initial drop in plate voltage, which brings it below E, backbiases
the diode, effectively removing E from any further circuit control.
Rundown and bottoming proceed as before, to the same final values,
except that the starting point is from a lower initial voltage and the sweep
ends sooner.
During the recovery of the plate toward Ebb, the diode conducts
when E b reaches E, stopping the recharging of C; the recovery time is
reduced to some percentage of its pre
vious value. And often, for just this
important reason, a diode is connected
10 K between the plate and a voltage divider
from Ebb to ground.
+300
500 K
20 K
Fig. 612. Millersweep circuit for
Example 61.
Example 61. Figure 612 shows a Miller
sweep circuit which incorporates a plate
bottoming diode, biased to conduct when
the plate falls to 200 volts. Thus the
duration of the linear sweep depends on the
plate rundown rather than on the grid
voltage rise. Furthermore, in this circuit the
source impedance is relatively large and
cannot be neglected.
The initial value of the grid voltage is
£c(0) 
1 K
500 K
300  0.6 volt
After switching into the active region, A — —50, r„  Ri, = 50 K, and the reflected
source impedance is 1 K. From Eq. (620)
£,(0+)
ii + 5W^ 300 =  2  34volt8
which is well above the cutoff value of —3 volts. The total grid change is —2.94
volts, and the plate drops by the same amount.
From Eq. (624),
e b (t) = 297  0.6 X 10«i
The plate runs down 97 volts to the 200 volts at which the diode conducts in
97
h  <T6 X 10 "
During this interval, the grid rises by
AE, =
to
Aff t 97
A = 50
e«(«i) = 0.4 volt
162fisec
1.94 volts
66. Recoverytime Improvement. The long recovery time is pri
marily a function of the large plate load of the Millersweep tube. If
Sec. 65]
VACUUMTUBE VOLTAGE SWEEPS
187
in the capacitor charge and discharge paths we could somehow replace
this large source impedance by a much smaller value without otherwise
affecting the circuit, then the recovery time would be reduced proportion
ately. One convenient method is to isolate the capacitor from the plate
circuit by means of an intervening cathode follower (Fig. 613a).
A cathode follower's gain, especially when using a highjj tube and a
large value of R K , is almost unity. Since AcfEi, = E b , the cathode
follower couples everything happening at the plate of Ti to the capacitor
through its very low output impedance (Fig. 613b). The overall
circuit is identical with those previously given in Figs. 67 to 69, except
*1 ,, *2 "2
^r
I — WV — « ( » vw
=•£«
A CF E b
R*<
(a) (b)
Fig. 613. (a) Miller sweep with a cathode follower included for fast recovery; (b)
equivalent grid circuit of the sweep tube.
that the source impedance must be changed from Rl or R L  r vt as the
case may be, to
i — v kk = — 7t
M2 + 1 M2 + 1
R s2 is quite small: in the normal triode it would lie between 200 and 500
ohms. If we compare this small resistance with the 10 to 100K source
impedance that was formerly present, we can see that the bottoming and
recovery time constants (n and r z ) will be greatly reduced. Conditions
during the circuitvoltage jumps, rundown, and recovery remain as
before, except for the introduction of i? s2 in all pertinent equations.
The isolation of C from the plate of the Millersweep tube allows
extremely rapid recovery of the plate, limited only by any stray circuit
capacity from point b of the switch to ground. It is completely inde
pendent of the new Miller grid recovery time constant of [from Eq.
(629)]
r ??— ^ (631)
A
(' + 5 ! £l) c
188
TIMING
[Chap. 6
Assuming R, 2 = 500 ohms in the circuit of Fig. 613, this time constant
is reduced from 50 to 0.9 /*sec. The new waveshapes are sketched in
Fig. 614. Note that the rundown time constant [Eq. (621)] remains
relatively unaffected by the circuit change. But bottoming occurs
slightly faster, once the grid reaches zero, because of the reduction of t 2)
which was previously given by Eqs. (627). Just before opening the plate
switch, the charge on C was approximately #&,„»„; just after the switch is
opened the equivalent generator of the cathode follower jumps to E.
JE c (0)
Fig. 614. Millersweep waveshapes when a cathode follower is used for fast recovery.
The change of circuit voltage divides proportionately across r c i and
Tpn/iia + 1), with the new grid voltage becoming
■Ee.max =
Td
r.i + r p2 /(M2 + 1)
(E — Ei, min )
(632)
Equation (632) indicates that upon recovery a very large positive grid
voltage pulse will appear. In an actual circuit, the peak will always
be appreciably less than the value calculated above. Both the grid
resistance, which decreases markedly at higher values of grid voltage,
and any stray capacity present act to limit the maximum grid jump to
10 to 20 volts rather than the 50 or 100 volts calculated. This portion
of the cycle appears after the region of major interest (the linear run
down), and therefore we can accept a very gross approximation for the
solution. It still explains, as well as necessary, the waveshape that will be
seen on an oscilloscope.
The cathode follower also performs an important secondary function
by making the sweep voltage available at a low impedance level for
Sec. 66]
VACUUMTUBE VOLTAGE SWEEPS
189
coupling to other circuits. We can tolerate a much greater degree of
external loading at the cathode, without affecting the sweep, than could
possibly be applied at the plate of the sweep tube.
66. Sweep Switching Problems. The final sweep essential is the
establishment of the capacitor discharge path which terminates each
sweep cycle. Looking backward to Chap. 5, this function was served by
the gas tube. However, any other bistate device, such as the controlled
gates of Chaps. 3 and 4, might serve equally well. These circuits require
an external control signal which opens or closes the gate at the proper
point of the cycle and for a given time interval. But in reality the
extra signal involves slight, if any, additional complexity. In order to
stabilize the gastube sweep, we were also forced to inject an external
voltage, the sync signal. Thus the gate control voltage might simul
taneously perform two functions, switching and synchronization.
Fig. 615. Diodebridge sweep discharge circuit.
Suppose we concentrate our attention on only one of the possible
switches, the diode bridge of Sec. 36, and apply it to the three sweeps
considered so far in this chapter. Each has its own special capacitor
discharge or charge problems, and in the aggregate they adequately repre
sent what might be expected in almost any switching situation.
First consider the current feedback sweep of Sec. 62 (Fig. 61). The
diode bridge, applied across the capacity (Fig. 615), should normally
be open: all diodes should be backbiased (e„ negative). Unless the
diode's reverse resistance is very large, the shunting of C by r r will
adversely affect both sweep time and linearity. In addition, the diode
capabilities limit the maximum sweep amplitude to somewhat less than
twice the peak inverse voltage. The control pulse forces the bridge
into conduction, shunting C with r t and thus discharging C. For com
plete discharge the controlpulse duration must be at least Ar f C. Pro
vided that the spacing between adjacent pulses is less than or equal
to the maximum linear sweep time [Eq. (65)], the periodic discharge
locks the sweep time to the controlpulse period.
190 timing [Chap. 6
The diode bridge may also be used in shunt with the capacitor of the
bootstrap sweep of Sec. 63. Amplitude limitations, required control
pulse duration, and the mechanism of switching are identical with that
of the current feedback sweep. However, in this sweep we can compen
sate for the shunting effect of r, which had previously increased the sweep
nonlinearity. The input impedance R/{\ — A) is made negative and
equal in value to r T by making the base amplifier gain slightly greater
than unity. The parallelresistance combination becomes infinite, ensur
ing constantcurrent charging of C, even in the presence of the diode
bridge.
In the triode Miller sweep (Sec. 64), the switching circuit and recovery
path are completely independent. The bridge, inserted in series with
the plate, simply replaces the switch of Fig. 67a. When it is pulsed
to cutoff (diodes backbiased), the large reverse resistance r r in series
with the plate greatly reduces the loop gain and effectively opens the
plate circuit, allowing recovery. When the bridge again conducts, the
next sweep cycle starts. An alternative is to shunt R L with a normally
conducting bridge and by this means reduce the load resistance, and hence
the gain, to zero. Once the pulse opens the gate, the change in resistance
produces the initial grid drop, with the amplification, now permitted,
starting the linear rundown.
The conclusion which can be drawn from the above arguments is
that almost any means of discharging the energystorage element, inter
rupting the feedback path, or reducing the loop amplification to zero will
satisfactorily control the sweep operation. Of course, we must ensure
that the circuit enters its active region upon switching; if it remains
saturated or cut off, the sweep may be delayed or may never even get
started.
67. Pentode Miller Sweep. Rather than separate the functions of
gating and sweep into two independent circuits, they might well be
incorporated in a single tube, a pentode. The suppressor is normally
biased negatively enough to completely cut off the plate current. In
order to turn the plate on and so complete the feedback loop, we must
apply a positive control pulse at the suppressor. The subsequent circuit
operation, the equal drop in plate and grid voltage, the linear charging
and eventual bottoming, and finally the recovery are similar to those of the
triode Miller sweep. Furthermore, with its greater amplification, the
pentode improves the sweep linearity by a factor of 10 or more over that
of the triode and also permits a greater sweep amplitude.
The basic differences between the pentode Miller sweep of Fig. 616
and the triode sweep appear in all boundary conditions dependent on the
physical characteristics of the tube. Practically, this sweep would also
include a cathode follower for fast recovery and a plate catching diode,
Sec. 67]
VACUUMTUBE VOLTAGE SWEEPS
191
but in the interests of circuit simplicity, these elements have been
omitted from Fig. 616.
Under normal circuit operation, the initial conditions are the following.
The control grid, returned through R to £», conducts and therefore will
be slightly positive, just as in the triode. Since the suppressor cuts off
the plate, the cathode, control grid, and screen grid act as the three
elements of a triode. As a direct consequence of the positive control
grid, heavy screen current flows, with the resultant voltage drop in R e
lowering the screen voltage to 50 or 60 volts.
Once the positive gating pulse raises the suppressor voltage to, or
more usually slightly above, zero, plate current will begin flowing. In
order to maintain the gating voltage substantially constant over the
sweep cycle, the control pulse must be applied either through the very
long time constant C,R, or directly from a dc source. An argument
similar to the one employed with
the triode sweep leads to the same
conclusion; upon switching on, the
plate and control grid both drop
by an amount not quite sufficient
to cut off the tube. Calculation
of the exact initial drop (about 10
volts) is not as simple a process as
previously described, because the
screen characteristics now have the
predominant control over the tube
current (Sec. 69). The low grid
voltage reduces the total pentode
current almost to the vanishing point, and therefore the screen voltage
simultaneously jumps to nearly Em.
After the initial jumps, the grid starts rising toward £&& at a rate
given by Eq. (623), beginning the corresponding plate rundown. The
large time constant t\ is identical with that of the triode sweep [Eq.
(622)]. The initial conditions, voltage jumps, linear rise, and recovery
are illustrated in the sketch of the circuit waveshapes (Fig. 618), and
for clarity we should periodically refer to them during the following
discussion.
The next question facing us is, Where does the sweep end? The higher
pentode gain (A = 250) permits the plate to fall to about zero while the
grid rises by only 1 volt, or even less, from its approximate initial value
of — 10 volts. Consequently, the limits of sweep operation are no longer
determined by the grid but depend instead on the plate's bottoming.
Figure 617 shows a portion of pentode plate voltampere characteristics
represented by a set of straight lines. The large value of load resistance
t 2 1
Fig. 616. Pentode Miller sweep.
192 timing [Chap. 6
Rl was, of course, chosen for the highest possible amplification, and
therefore the load line will be almost horizontal.
Note that no further voltage drop is possible after the plate falls to
the point where the load line intersects the knee of the characteristics,
Eb. mm At this operating point the plate resistance changes to a very low
value, about 10,000 ohms, and g m is reduced to zero. Since the circuit
amplification also becomes zero, the Miller effect ceases. Plate bottom
ing occurs at 2 to 10 volts, depending upon the characteristics of the
particular tube and the load resistance chosen. This value is so very
Fig. 617. Millersweep tube — plate characteristics and load line.
small in terms of the total sweep amplitude that it is often approximated
as zero. A somewhat better approximation would be
E b
> = Rl Evb
where r'„, the effective plate resistance, is the reciprocal of the slope of the
«i.,mi» line.
Since the grid voltage of the pentode remains almost constant dur
ing the linear plate rundown, the capacitor charging current is best
represented by 7 C (0+). By substituting this current, instead of Eu/R,
the necessary time for the complete plate rundown, which starts at
En, + E cl (0+) and ends at E b , mia , is
RC
E bb + E cl (0+)  E b „
Em,  E cl (0+)
RC
AE b
E^  E cl (0+)
(633)
During this same interval, the grid rises by only AE b /\A\ volts, where
AE b is the total change in the plate voltage. If the maximum plate
Sec. 67]
VACUUMTUBE VOLTAGE SWEEPS
193
voltage is limited to an external control voltage E by a plate catching
diode, the starting point of the rundown would be E + E cl (0+) instead
of En + E c i(0 + ). As in the triode circuit, the sweep duration is a linear
function of this control signal.
In Eq. (633), the only terms dependent on the tube are E c i(0+) and
Eb, a >n. These are directly proportional to Ebb, and therefore any small
decrease or increase in the supply voltage would change both the numer
ator and denominator proportionately. Thus we conclude that both this
sweep and the phantastron (Sec. 68) have excellent time stability with
respect to powersupply variations.
Ec max'
Fig. 618. Pentode Millersweep waveshapes.
Furthermore, the terms in Eq. (633) dependent on the tube, # c i(0+)
and Eb. min , are both small compared with Em,. By making R L large,
Et.min is reduced almost to the vanishing point. The other term, E c i(0 + ),
remains relatively constant over the life of the tube and does not change
much from tube to tube. To a very good approximation we can say that
this sweep is virtually independent of the tube. This is one of the major
advantages negative feedback offers over the positive feedback employed
in the bootstrap sweep.
Even after the plate bottoms, the grid continues charging toward En,
but very much faster, with a time constant reduced from r x = \A\RC to
r£' = C(R + r') S CR
194 timing [Chap. 6
The grid, charging from only a few volts negative, reaches zero in a
comparatively short time and begins conducting. All circuit conditions
again change; the time constant becomes
r4" = (r„ + r' v )G
and the steadystate voltage changes back to its initial value E c i(0~).
But the rapidly rising grid voltage increases the total cathode current,
almost all of which now flows to the screen. Therefore the screen
voltage will drop to slightly above its initial value, the small amount of
current flowing to the plate accounting for the difference.
Immediately upon the removal of the suppressor gating pulse, the
plate cuts off and recovery proceeds as in the triode sweep, with all the
voltage jumps and time constants found in a similar manner. The only
additional consideration is that the positive grid jump also appears as an
amplified drop at the screen.
The pentode circuit, besides producing the linear plate rundown,
simultaneously generates a large rectangular pulse of equal duration
which has fast rise and fall times. This signal, which appears at the
screen, is as important as the linear sweep. We might apply it to the
normally cutoff grid of a cathoderay tube, thus unblanking the oscillo
scope only during the linear sweep (supplied by the plate rundown).
Since its period is well defined, it can be used in conjunction with a gate
for accurate time selection. If differentiated, the negative output trigger,
which is delayed from the initial positive trigger by the sweep period,
serves either for timing measurements or to start subsequent operations.
A few words are now in order concerning the special requirements
we impose on the pentode used in this sweep. First, it should have sharp
suppressor control over plate current flow to ensure definitive onoff
circuit states. Secondly, since under platecurrent cutoff conditions the
total cathode current flows to the screen, the tube should have the
capabilities necessary to dissipate the heat produced ; its maximum screen
dissipation must be large. Only slightly less important is a small control
gridscreengrid transconductance to ensure that the screen voltage will
remain reasonably constant during the linear grid rise. Otherwise the
screen degeneration reduces the effective platecircuit amplification and
thus increases the sweep nonlinearity. Special tubes, such as the 6AS6
and the 6BH6, whose specifications satisfy the above requirements, have
been developed primarily for Millersweep and phantastron applications.
68. The Phantastron. For optimum circuit response, the ideal
Millersweep gating pulse would be one having exactly the same duration
as the linear plate rundown. The circuit would recover immediately
upon the plate's bottoming, becoming ready, in the shortest possible time
interval, to react to the next input pulse. But the Miller sweep itself
Sec. 68] vacuumtube voltage sweeps 195
generates a pulse of just the proper duration, and therefore we might
just as well let the circuit do its own gating. All that this requires is
the coupling of the screen pulse directly into the suppressor. Since the
two waveshapes, the sweep and gating pulse, are simultaneously pro
duced, we are not forced to cope with the problems of synchronization
and phasing that we would be sure to meet in attempting to generate an
independent control signal.
Figure 619o shows the complete phantastron circuit including a
cathode follower for fast recovery and a plate catching diode D\ for
voltage control of sweep duration. We normally adjust the coupling
network R c , R a , and R, to keep the suppressor at about —20 volts, i.e.,
sufficiently negative to ensure plate cutoff. And as in the Miller sweep,
the screen conducts heavily, setting the quiescent voltage level at 50 or
60 volts.
Application of a narrow positive trigger at either the suppressor or
the screen starts the sweep by raising the suppressor voltage to zero or
even slightly higher, thus bringing the plate out of cutoff. The resultant
drop in controlgrid voltage almost cuts off the entire tube, and conse
quently the screen will jump toward Em. This jump, coupled through
C c and the resistor network, keeps the suppressor turned on, even after
the starting trigger disappears. C c serves to speed up the switching
action by immediately coupling the sharp rise and fall of the screen
voltage into the suppressor, thus counteracting the retarding effects of
the stray circuit capacity. It functions in a manner similar to the capaci
tor in a compensated attenuator which is adjusted somewhat overcom
pensated; usually C e is very small, only 25 to 100 nnf.
Diode Z>2 limits the maximum positive suppressor voltage to about
5 volts, set by the voltage divider Ri and fl 2 . We pick this voltage to
give the largest possible gain, since g m reaches its maximum value at a
slightly positive suppressor voltage. In addition, the suppressor current
increases with increasing voltage, and unless limited to a very low value,
the power dissipation may exceed the tube's ratings.
Until the plate bottoms, the sweep operation is identical with that of
the Miller sweep of Sec. 67, taking the same time to generate the same
waveshapes (Fig. 619). However, after the plate bottoms, the drop
in screen voltage is directly coupled to the suppressor and turns the plate
off. The cathode follower allows fast plate recovery to its initial value
E and also Contributes a large positive grid jump, which, as we expect,
appears amplified at the screen. But this additional drop can only
help to turn the suppressor off even faster.
The gridcircuit recovery is extremely rapid, because of the cathode
follower and the controlgrid conduction path. And the system is now
ready for the next input trigger.
196
TIMING
[Chap. 6
We may inject the positive trigger pulse at either the screen or sup
pressor, the speedup capacitor effectively applying it at both elements
simultaneously. The trigger not only starts the suppressor into the
plate conduction region, but also aids the initial screen rise. Its ampli
tude and duration are not critical, provided, however, that it at least
exceeds the threshold necessary for guaranteed switching.
Trigger
*C2
e C 3
£»,
£ c2 (o)
t^c max
. E cl (0)
AAi 1 ^
/
.
_______^
h
t
<T>
t
t
k ■
h.
Fig. 619. Phantastron circuit and waveshapes.
Example 62. The phantastron of Fig. 619 employs a 6AS6 with the plate load
adjusted to give a gain of 250. We can assume that initially the grid drops to — 10
volts and that the plate finally bottoms at 5 volts. Furthermore, in this circuit,
Ebb = 300 volts and E = 200 volts.
(o) What RC product is required to give a sweep time of 500 /isec?
(6) Under these circumstances, what is the sweep nonlinearity?
(c) Assuming that i? c i(0 + ) changes by 10 per cent, by what percentage would the
sweep duration change?
Sec. 68] vacuumtube voltage sweeps 197
Solution, (a) The total plate rundown is
AE b = 200  10  5 = 185 volts
From Eq. (633),
RC = h £w, .j; 10 = 500 X 10' X ~, = 825 X lO'sec
loo loO
Let R = 1.65 megohms and C = 500 upd.
(b) The change in the grid voltage over the complete linear rundown is only
Thus the grid rises from — 10 to —9.26 volts out of a charging curve having a 300volt
maximum value. Substituting these limits into the linearity equation (522) yields
NL= 12.5% X§^jS0.03%
Actually, this value may be too small to have any significance. The nonlinearity of
C over this voltage range and the various secondorder effects would, at the very
least, double the calculated value.
(c) From Eq. (633), with E.i(0 + ) = 9 volts,
The normal sweep is
t\ = RC X 1 8 %09
U = RC X ls Hio
Thus, by dividing and expanding,
t\ /l86\/310\ (185 + 1)(309 + 1) , 496
= 1 + ,.„,,,. = 1.0087
l l = (—\ (—\ 
h ~ \185/ \309/ ~
h \185/ \309/ (185) (309) ~ (185) (309)
The sweep time is increased by only 0.87 per cent when the initial grid drop is reduced
by 10 per cent.
Freerunning Phantastron. The phantastron lends itself to selftrig
gering or freerunning operation. By simply setting the quiescent sup
pressor bias within its base, the plate is normally conducting. Consider
the circuit behavior immediately following plate bottoming. The screen
voltage drop, together with the large sharp spike (contributed by the
control grid's positive jump), momentarily drives the suppressor below
plate cutoff. As the control grid recovers, both the screen and suppressor
voltage follow it toward their quiescent values. But the suppressor
eventually reaches a point where the plate can turn back on, and as a
consequence, the switching cycle repeats. The screen, driven back
toward En, pulls the suppressor up along with it. Rundown begins
again, and the cycle keeps repeating.
Synchronization of the phantastron is usually effected by converting
the input signal to pulses, which are then used to turn the sweep on or
maybe to turn it off. These control either the start or the sweep bottom
198
TIMING
[Chap. 6
ing point, and the resultant regions of synchronization may be defined
in exactly the manner of Sec. 56.
69. Miller Sweep and Phantastron — Screen and Controlgrid Volt
age Calculations. To attempt an exact solution for the pentode screen
and controlgrid voltages is to attempt an extremely difficult task. We
would need a complete set of both plate and screen voltampere character
istics, and even then probably the best approach would be one of suc
cessive approximations, i.e., making a guess, checking it, and then making
a more educated guess, until finally some guess agrees with the checked
answer.
Fig. 620. Pentode screen circuit models, (a) Plate cut off; (fe) plate conducting.
However, we can quite simply find an approximate solution, by a
method which still keeps the essential circuit behavior in the forefront.
This is through treating the cathode, control grid, and screen as a triode.
It follows that the screen model of Fig. 620a represents the circuit
when the suppressor cuts off the plate and the total current flows to the
screen. Rth and E T h are actually the Thevenin equivalents of the screen
network. The screen parameters r cU and ju c2 are determined from the
tube under triode operating conditions, plate and screen connected
together so that i c i = ik
Mc2
de c i
Al.i0
Tcii =
did
de c i
dih
Aei0
(634)
The parameters of Eqs. (634) may be found directly from the manu
facturer's curves by adding the screen and plate current characteristics.
For the two widely used sweep tubes, reasonable values are :
Sweep tube
liel
r««, kilohms
p
6AS6
6BH6
25
20
5
4.5
3
2
Sec. 69] vacuumtube voltage sweeps 199
Using these values, and with e c \ = 0, the quiescent value of e c « may
readily be calculated from Fig. 620a.
tlTh T" ~cU
Once the suppressor pulse allows plate current flow, then its sole
role is to regulate the percentage of the total cathode current that flows
in the plate circuit. But the total current remains predominantly a
function of the screen and controlgrid voltages, and therefore the form
of the screen model is still consistent with the tube's physical behavior.
In the positive suppressor region, the division of current between the
plate and screen is almost constant and is independent of the suppressor
voltage. This ratio may be expressed as
p = A (636)
The constant p depends on tube geometry and is also found from the
manufacturer's curves; it may be taken as 3 for the 6AS6 and 2 for the
6BH6. When p is 3, only onequarter of the total cathode current flows
to the screen. With the tube now operating as a pentode,
ik = ib + id = (1 + p)»c2
and ictp = — ; — r
p + 1
Substituting into Eqs. (634) yields
r,i T = ^7— = (p + 1) 37 = (P + !)'«»
OlcZp Oik
Thus r c u must be multiplied by (p + 1) in order to account for the
effects of plate current flow and the screen model will be changed to
the one shown in Fig. 6206.
The equal drop in plate and controlgrid voltage, upon switching, may
be expressed solely in terms of the screen current by noting that
E eJ (0+) £i i b R L = piciRL (637)
where Rl is the effective plate load resistance. Substitution of Eq.
(637) into the model of Fig. 6206 establishes all the conditions necessary
to solve for E c i(fl + ). And it follows directly that
p , n+ \ pEthRl
° lK ' HczpRl +(p + 1>c2« + R™
But since r c n(p + 1) + Rth <K hcipRl,
Eth
E cl (0+) S*  — (638)
Mc2
200 timing [Chap. 6
Equation (638) is just the cutoff voltage of the screen circuit, treated as a
triode, which verifies our assumption that the tube is driven almost to
cutoff. At the same time the screen rises to nearly Eth, as might be
found from Fig. 6206.
During the controlgrid rise, the screen voltage droops slightly because
of the amplification from control grid to screen,
AcUi = ~ (p + l)rl"+ R Th (6  39)
In order to prevent excessive changes in the screen voltage, the ampli
fication must be kept small. To this end we usually choose R Th to be
of the same order of magnitude as r c2 «(p + 1). Therefore the gain is only
10 or 12. Since the controlgrid voltage changes by a fraction of a volt
over the whole sweep cycle, the screen voltage will remain substantially
constant.
Example 63. We shall now consider the design of the screensuppressor coupling
circuit for the phantastron of Example 62 (Fig. 619). The desired quiescent condi
tions are E ci = 50 volts and E c % — — 20 volts. The two supplies at our disposal are
+300 and 200 volts.
?+300
ScreenI 5Q
Suppressor* 20
200
Fro. 621. Model for Example 63.
Solution. The model holding under plate cutoff conditions is shown in Fig. 621.
We arbitrarily choose a 1ma bleeder current through R a and R,. If this current is too
large, appreciable power would be wasted; if too small, the resistors would become
excessively large. Therefore
70volts = 70K
1 ma
l^volts = lg0K
I ma
Since r c% , = 5 K, the screen current at 50 volts is 10 ma. Thus 11 nia must flow
through Rt, and
Be . 25p_, 23 K
II ma
VACUUMTUBE VOLTAGE SWEEPS 201
Prom these values, we can find the Thevenin equivalent of the screen circuit. It
is a 278volt source having an internal impedance of 22 K and 250 K in parallel, or,
to a good approximation, 20 K. The grid voltage after switching becomes, from
Eq. (638),
#.i(0 + ) =  21 %t S 11 volts
The gridtoscreen amplification is [Eq. (639)]
. _ 25 X 20 K _ 10 g
A cUi (3 + 1)5 K + 20 K
In the previous example the controlgrid voltage only changed by 0.74 volt. Conse
quently the screen voltage will run down by only 9 volts from its starting point
of approximately 278 volts. These values check quite closely with laboratory
measurements.
PROBLEMS
61. (a) Prove that varying E in the circuit of Fig. 61 is a very unsatisfactory
method of adjusting the sweep time. Consider the sweep linearity as E charges from
10 to 100 volts. The final voltage remains constant at Eu.mai
(6) Under the conditions of part a, how will the sweep duration change if the plate
is always allowed to bottom before the capacitor is discharged?
62. The sweep of Fig. 61 uses a 0.05mI capacitor for timing. Once the switch is
opened, it remains open. Sketch the plate waveshape if the grid resistance in the
positive grid region is 1 K. The model which should be used for the plate bottoming
region is a resistor of r„ = 700 ohms (no controlled source) from the plate to the
cathode. This model holds for e c > e& (Prob. 44).
63. Assume that we desire to construct a bootstrap sweep but that we are unable
to obtain an amplifier with a sufficiently stable gain A. Instead, we can make ft a
function of A, so that, within rather narrow limits, the change in ft will compensate
for the change in A and thus maintain the sweep time invariant. Find the required
functional relationship, that is, ft = f(A). Find the approximate relationship when
A is close to unity. Repeat if R = f(En).
64. The only tube available for the sweep of Fig. 61 has y. => 5 and r„ = 5 K.
The other circuit values are En = 300 volts, E — 100 volts, ft* = 50 K, and C =
0.002 4.
(a) Sketch the plate waveshape, giving all values if the switch is opened at t =
and closed when e„* reaches zero.
(b) Calculate the NL of this sweep.
(Hint: All the equations in Sec. 62 may not hold with a low^ tube.)
65. (a) Draw the capacitor and output waveshape on the same axis if A = 0.95,
E  100 volts, ft = 1 megohm, and C = 100 «rf in the circuit of Fig. 63. The
capacitor is discharged when its terminal voltage reaches 100 volts.
(b) Calculate the sweep NL.
(c) Compare the results of part a with the results obtained when A — 1. For this
comparison the capacitor is adjusted to maintain the same sweep period.
(d) This sweep is adjusted by varying E. Plot the time duration versus K if
E = 1/KE, (A = 1).
66. The bootstrap of Fig. 622 employs a cathode follower as its base amplifier,
(a) Sketch the voltage waveshape appearing at the cathode if the switch is opened
at t = and closed once the gridtocathode voltage reaches zero. Label all voltage.
202 timing [Chap. 6
values, time constants, and times. (Hint: Replace the tube by the equivalent circuit
seen when looking into the cathode.)
(6) Show that within its active region this circuit may be represented by the model
of Fig. 64. Specify the parameters.
(c) Calculate and compare the sweep amplitude and linearity when the battery in
series with the charging resistor is present and when it is absent.
+200 v
M 100
r„100K
f+300v
1 r„=50K
/ il^f r c =2K
Fio. 622
Fig. 623
67. The circuit of Fig. 623 makes use of the techniques discussed in Sec. 62 to
generate a specific nonlinear sweep. Sketch ei, e*, and e 3 to scale if both switches are
opened at t = and closed when the voltage across the tube, ebt, drops to 200 volts.
What function does this circuit generate? Make all reasonable approximations in
your solution. Justify any assumptions made.
68. In the Miller sweep of Fig. 67, the switch is closed at t = and opened a
short time after the plate falls to its lowest value. Sketch the plate and grid wave
shapes, giving the values of all voltages, times, and time constants. Compute the
sweep nonlinearity of the linear plate rundown. The tube and circuit parameters are
Rl = 240 K
R = 1 megohm
C = 1,000 «if
n = 100
r p = 50 K
r» = 2K
En = 300 volts
69. Repeat Prob. 68 if the plate is returned through a plate catching diode to
+200 volts. Pay particular attention to the time required for the linear plate run
down and for the time required for the complete plate recovery.
610. (a) Compare the sweep linearity of the triode Miller sweep under the follow
ing conditions:
1. The charging resistor is returned to 2?t*.
2. The charging resistor is returned to a voltage equal to O.lEu,. Express the
answer as a ratio (assume \A\ 5J> 1).
(b) Prove that the sweep period is a linear function of the control voltage E.
(c) Prove that the sweep period varies inversely with the voltage to which the grid
resistor is returned.
611. Figure 624 represents a variation in switching the Miller sweep on and off.
Sketch the grid and plate waveshapes, labeling all times and break voltages. (Hint:
Be careful in evaluating the initial conditions of each region.) The switch is opened
at t = and closed soon after D t conducts. What is the largest voltage which can
VACUUMTUBE VOLTAGE SWEEPS
203
be used to backbias the tube and still allow the sweep to start as soon as the switch
is opened?
9+300 v
I100K
2M
500 w*
)Y
w
D 2
€)
100 v
M =50
r p =25K
r c =500
Fig. 624
612. (a) Compute the initial grid drop, the time required for the linear plate run
down, and the recovery time in the improved sweep of Fig. 613a. The circuit com
ponents are
Eu = 300 volts R = 1 megohm in = 20
E = 0.75£» R k = 40 K r r! = 10 K
Rl = 200 K m = 100
C = 1 Mf r pl = 100 K
Assume that the total stray capacity from the plate of 2\ to ground is 50 «if
(b) Sketch the waveshape at the cathode of the cathode follower and calculate the
internal impedance at this point.
(c) What is the maximum possible value of E before the grid of the cathode follower
is forced into conduction? If E exceeds this voltage, what happens to the circuit
response?
613. The sweep of Fig. 625 is placed in operation by opening the switch <S at
t = 0. It is closed again at t = 200 jusec. Plot the grid and plate voltage to scale
from before t = until the circuit completely recovers.
g m = 1,000 /jmhos
/j100
c 1,000 mi\
200v=
=100v
Fig. 625
614. Show three methods of switching the triode Miller sweep which are adaptable
to diode or triode gating circuits. Discuss any limitations on or modifications in the
basic sweep behavior when each gate is inserted. Give the circuit of these gates
together with their points of insertion and the gating signal requirements.
615. In the phantastron circuit of Fig. 626, assume that the plate falls by 10 volts
when the pulse is injected. The plate rundown ends when D, conducts. If the
204
TIMING
[Chap. 6
loop gain is assumed to be —200, sketch the plate and grid waveshapes. Label
these plots with all voltage and time values. Calculate the sweep nonlinearity.
9+300
p=2.S
r„~500 K
g m  1,000 /imhos
r c2 =10K
^ c2 =20
r,,lK
Trigger
D input
4+20 A 100
Fig. 626
616. Plot the correct suppressor, screen, plate, and controlgrid waveshapes for
the circuit given in Fig. 626. Assume that screen degeneration results in a 40 per
cent decrease in the controlgrid to platecircuit gain. Give all voltage values, times,
and time constants.
617. The circuit of Fig. 626 has been modified to the circuit of Fig. 619. Repeat
the calculations of Prob. 615 if the additional circuit parameters are
Triode n = 100
r„ = 100 K
R k = 200 K
E = 200 volts
Ri = 100 K
R* = 5K
Make all reasonable approximations in your calculations and assume that the stray
capacity loading the pentode elements is completely negligible.
618. A phantastron used as a linear sweep is shown in Fig. 627. As a first
approximation assume:
1. Cathodefollower gain ^ 1
2. An initial drop of 10 volts at the plate
3. r v = 525 K and g m = 2,000 jimhos during rundown
Given the cathode voltage waveform as shown in Fig. 627, calculate the approxi
mate plate waveform for the first 300 iisec after a trigger is applied. At the end of
this interval the tube is turned off. If the total capacitance from the phantastron
plate to ground is 100 nrf, calculate the flyback time.
p+300v
200 v=
Fig. 627
VACUUMTUBE VOLTAGE SWEEPS
205
619. The phantastron circuit of Fig. 619 is adjusted so that the normal suppressor
voltage is zero (freerunning sweep).
(a) Discuss the effects on the sweep waveshape of injecting a synchronizing signal
into the grid, screen grid, suppressor, and plate. This signal consists of a pulse train
of equally spaced positive and negative pulses with a spacing between adjacent pulses
of 75 /isec. The phantastron freerunning sweep period is 1,000 jusec. Which point
would be the best place to synchronize the sweep?
(b) Plot the regions of pulse synchroni
zation if the sync signal is applied as 9+300
shown in Fig. 628.
+300
esync
Fw. 628
620. This problem is designed to investigate the region of freerunning phantastron
operation between the time that the plate bottoms and the time that the next sweep
starts. We do not have to consider the sweep rundown, but can concentrate atten
tion on the screen and suppressor coupling. Suppose that a 6BH6 is used in this cir
cuit, biased at E e i = 100 volts and E cl = 40 volts when the plate circuit is opened
and when Di is removed. With the plate supply used, the suppressor cutoff may be
taken as — 15 volts. Assume that the positive grid jump drives the screen down to
10 volts, from where it recovers with the grid time constant of 2 i«sec. The plate and
grid are decoupled by a cathode follower.
(a) Calculate the value of resistors in the bleeder network of Fig. 629 needed for
proper biasing.
(6) Sketch the screen and suppressor waveshapes, giving all voltage values and
times. Assume that the plate starts at 300 volts and runs down (after the initial
drop) to zero in 50 jusec.
BIBLIOGRAPHY
Briggs, B. H.: The Miller Integrator, Electronic Eng., vol. 20, pp. 243247, August,
1948; pp. 279284, September, 1948; pp. 325330, October, 1948.
Chance, B.: Some Precision Circuit Techniques Used in Waveform Generation and
Measurement, Rev. Sci. Instr., vol. 17, p. 396, October, 1946.
et al. : "Waveforms," Massachusetts Institute of Technology Radiation
Laboratory Series, vol. 19, McGrawHill Book Company, Inc., New York, 1949.
Close, R. N., and M. T. Kibenbaum: Design of Phantastron Time Delay Circuits,
Electronics, vol. 21, no. 4, pp. 100107, 1948.
Puckle, O. S.: "Time Bases," 2d ed., John Wiley & Sons, Inc., New York, 1951.
Williams, F. C, and N. F. Moody: Ranging Circuits, Linear Time Base Generators
and Associated Circuits, /. IEE (London), pt. IIIA, vol. 93, no. 7, pp. 11881198,
1946.
CHAPTER 7
LINEAR TRANSISTOR VOLTAGE SWEEPS
We rightly expect that almost all the basic sweeps of Chap. 6 may be
adapted for transistor operation. Of course, the reverse transmission
path present within the transistor and the vast difference in impedance
levels do not permit its automatic substitution for the vacuum tube.
But if the required minor circuit modifications are made, a transistor
will perform at least as well as a triode in many of the voltage sweeps,
and even somewhat better in some of them.
Throughout the following discussion the reader should refer back
to the appropriate sections of Chap. 6, both to review the basic concepts
of the activeelement sweeps and as a means of recognizing the differences
between the transistor and vacuumtube circuits. Even though the
fundamental defining equations may be the same, it is these very differ
ences which account for the proper operation of each sweep. However,
the basic similarity that does in fact exist broadens our outlook by
enabling us to separate the system's behavior from the individually
chosen components. If the same circuit operates with either a triode
or a transistor, what is to prevent it from also working when some other
active element is substituted?
71. Constantchargingcurrent Voltage Sweep. One of the simplest
possible voltage sweeps makes use of the current source of the transistor
for the linear charging of the sweep capacitor (Fig. 71). This circuit's
operation is analogous to that discussed in Sec. 62, but as we note, the
almost constant current output makes the introduction of additional
current feedback unnecessary for many applications. Furthermore, since
the collector current is controlled by the emitter input, a means of
adjusting the charging rate is afforded at a terminal well removed from
the sweep output.
The approximate model given in Fig. 716 adequately represents
the circuit behavior because the emitter and base resistances are so
very small compared with the external controlling resistance R that
their neglect will have no appreciable effect on the operation. From
this model we see that the essentially constant emitter current is given by
L = §< (71)
206
Sec. 71]
LINEAR TRANSISTOR VOLTAGE SWEEPS
207
where E cc is the emitter bias source. After the switch is opened, the
collector charges from its initial value of E» toward the Thevenin steady
state voltage of
Eut = —aleTe = —
aE a
R
with the long time constant t\ = r c C.
But in order to have a reasonably large value of current flow, R will
have to be relatively small, i.e., no greater than several thousand ohms.
Since r c is normally greater than 1.0 megohm, the output apparently
charges toward a very large negative voltage. Once the collector drops
to zero, the transistor saturates with the now conducting collector diode,
nsntn
~Eu
(a)
VW — °— M
<fb
e
j
(b)
Fig. 71. (o) ConstanfrKjurrent sweep and output waveshape; (6) activeregion model.
shorting the capacitor to ground. Only a very small percentage of the
total exponential appears at the output, and therefore it seems reason
able to approximate this voltage by an absolutely linear rundown.
e,(t)
E»jjfUdt = Eu,aj£t (72)
Substitution of the bottoming value of zero into Eq. (72) yields a maxi
mum sweep duration of
RC
tE c ,
(73)
Equation (73) points up the dependency of this sweep on the value of a
and hence the necessity of recalibration upon the replacement of the
individual transistor.
208
TIMING
[Chap. 7
Referring to Eq. (73), we see that since a low value of resistance was
used in the emitter circuit in an effort to maintain linearity, we are
forced to turn to relatively large capacity for any given sweep dura
tion. The use of high capacitor values is one of the characteristics
of transistor sweep circuits that differentiates them from their vacuum
tube equivalents.
If this sawtooth is intended to drive a following transistor stage, the
relatively low input impedance to be expected may load the capacitor
excessively, with a corresponding deterioration in sweep quality.
Zf
— Ecr
==c
c 2
,,e.
2 e.
1
n
Fig. 72. Transistor switching of the sweep circuit.
Switching in the circuit of Fig. 71 may be accomplished by shunting
the sweep capacitor by the complementary pnp transistor shown in
Fig. 72. Under normal conditions the additional transistor T 2 conducts
heavily and shunts C with its low saturation resistance. Injection of a
positive pulse through the input coupling capacitor C 2 raises the base
voltage of T 2 above E bb and rapidly cuts off the switching transistor.
During the presence of the pulse, the very high back impedance of T 2
will not prevent the expected linear rundown. Of course, this switching
pulse should be long enough to allow bottoming of the sweep.
Example 71. The recovery of the sweep of Fig. 72 is effected by recharging C
from an almost constant current source. This current must be of the opposite
polarity to that used for the original linear charging, and it is supplied from the com
plementary pnp transistor. The amplitude of the recharge current, and hence the
time required for recovery, depends directly on the size of Rl Consider a circuit
where £w> = \E C c\ = 10 volts, R = 1 K, and ai = ai = 0.98 and where the recovery
IK
:=l10v
+
Of (el
o
6,
'« C
— dfr
?P2
10v=
J '
Fig. 73. Model of the circuit of Fig. 72 holding in the recovery region.
Sec. 72] linear transistor voltage sweeps 209
time must not exceed 5 per cent of the linear rundown. Under these conditions,
what is the limiting size of iJj?
Solution. To a good approximation, the circuit of Fig. 73 represents the behavior
of the sweep immediately after Ti is switched back on. Both transistors are in their
active regions, and C is fully charged. Moreover, in approximating this circuit's
response, the small emitter input resistance is assumed to be zero and the shunting
collector resistance is assumed to be too large to influence either the sweep or recovery
times. During the linear, sweep, T* is off and the total current flowing into C is
%,i = — ort.i = — a Yrr = ~ ».e ma
During the discharge interval, the capacitor current can be found by writing the node
equation at point A :
i fl t — <xi,z — ai,i ™ ai 2 — 9.8 ma
E u 10
where
(1  a)Ri 0.02i£j
Since the charge and discharge of C are always from constantcurrent sources, the
ratio of times is inversely proportional to the current flow. Therefore, under the
conditions of the problem, for the recovery time to be 5 per cent of the sweep time,
»,! = 20i,i
Or to satisfy this condition,
t,« = 2li.i = 210 ma
Substituting and solving for Ri yields
B * ~ h R n g *\g = 2  375 ohms
i\ (.1 — a)CJc<:
Once C is completely discharged, current can no longer flow into it. To do so
would forwardbias the collector of Ti, and this would shunt C with the two conduct
ing diodes of the transistor.
72. Bootstrap Voltage Sweep. Bootstrapping entails positive feed
back of the sweep voltage around an amplifier having essentially unity
voltage gain. By this method we endeavor to maintain a constant
voltage drop across, and consequently a constant current flow through,
the charging resistor. One transistor bootstrap, illustrated in Fig. 74,
employs an emitter follower as its base amplifier. Over its complete
active region, the output is almost exactly equal to the applied base
voltage (the drop from the base to emitter is quite small). However,
the small voltage difference that does exist is inadequate to ensure
sufficient charging current flow, and therefore an additional battery must
be inserted in series with the feedback resistor.
Our starting point in the analysis of this emitter follower is the selection
of the proper model for the transistor. By choosing one based on Fig.
423, rather than a T model, the circuit is reduced to the twonode
network of Fig. 746, where, in addition, the feedback network (R and E)
was replaced by its Norton equivalent. Furthermore, the large collector
resistance r d is essentially in parallel with the much smaller R,, and there
fore it may be neglected.
210
TIMING
[Chap. 7
Since the transistor is a currentcontrolled device, it is more informa
tive to consider any variation in the circuit current instead of the resultant
change in voltage. The input controlling current i\ divides between the
external baseemitter resistance R and the transistor internal input
impedance r' u . The actual base input is
R
it
;ii
(74)
r'n + R
where r' n may also be expressed in terms of the T parameters as
r' n = n + r„{\ + 0)
By substituting Eq. (74) into the model, the controlled source may be
written as /3'i'i instead of as /34. Here
R
£' =
r' n + R
(75)
As a result, the circuit of Fig. 746 may be replaced by the simplified
model shown in Fig. 75.
(a)
(b)
Fig. 74. (o) Transistor bootstrap sweep circuit; (6) model holding within the active
region.
With the switch closed, the transistor must be in its active region.
It cannot be cut off because the supply current I flowing through R and
R, produces a voltage drop which forwardbiases the emitter base diode.
The current through R e in the active region is found from the model of
Fig. 75:
i a = (1 + /3')*'i  /
But the drop across R  r'n must also equal the drop across R e
EtQ) = [(1 + P')ii(0)  I]R. = ii(0)R  r' u
Solving this equation yields
*i(0) =
IRe
R II r'n + (1 + /?')««
Hence
(76)
Sec. 72]
LINEAR TBANSISTOB VOLTAGE SWEEPS
211
Once the switch is opened, the current that formerly bypassed C
begins charging it. The circuit is not disturbed when the sweep starts,
and no voltage jumps appear. Thus the initial value of the charging
current is
*,(0+) = /  ti(O) =
In general, 8'jR e S> R  r'„ and
P
i,(0+) S :
7 I (77)
R [ rjx + PR,
R II r' u + (1 + P)R.
R\\r' u
«'i., J_ (1 + P')h
V
I
o
fix
■5
l S e
t
*2
1+/3'
We might now note that almost
the complete bias current goes to
charge the capacitor. The remain
ing small amount sustains the
proper operating point of the
transistor.
Charging continues until the
transistor saturates. But the point
at which this occurs is known; it is when the drop across R, rises to E».
The corresponding limit of ij. is found from
E 2s = E a = [i u {\ + p)  I]R e
I R e + Em
Fig. 75. Simplified model of the tran
sistor bootstrap of Fig. 74.
lu
(1 + &)R.
Referring back to the input node, the final value of charging current in
the linear portion of the sweep becomes
f,(*i) = /
%\. =
P
1 +p
,1
En
(78)
(1 + P)R,
The second term in Eq. (78) is the change in current over the sweep
interval, and for maximum linearity it must be small compared with the
initial value. From Eqs. (77) and (78),
PI
p p >>_ p~
K ti e
For the purposes of comparison, we shall set E  E M . By also sub
stituting the value of p given in Eq. (75), the necessary inequality
reduces to
PR. » r'„ + R (79)
We conclude that R should be small and that a transistor with a large
should be used if acceptable linearity is to be achieved. Usually R,
limited solely by the current capabilities of the bias source, would be
212 timing [Chap. 7
on the same order of magnitude as r' u , i.e., about 200 to 1,000 ohms.
With the /S's available of 60 to 100 or higher, the inequality of Eq. (79)
is not difficult to satisfy with reasonable values of R t .
Example 72. Suppose that /3 = 50, R = r' u = 1,000 ohms, R, = 2,000 ohms, and
Et>b = E = 10 volts. Under these circumstances /3' = p/2 = 25 (from Eq. 75).
The initial charging current, given by Eq. (77), is
^ 0+) = 2m x wo = 9  62ma
This is only 4 per cent less than /. At saturation the charging current is reduced to
^= 9  62 26(P00T = 9  43ma
The total change of current is only 2 per cent, and linear charging of C appears to be
an acceptable approximation. Actually, the sweep is exponential, with the current
decreasing toward zero.
From Eq. (522), the sweep nonlinearity may be expressed as
NL = 12.5% 962 9 7 )2 9 ' 43 = 0.26%
Various secondorder effects which have been neglected in this discussion, such as the
decrease in 3 as the transistor enters the saturation region, may even increase the NL
by a factor of 2.
Since the preceding argument proves that sweep nonlinearity is small,
the input voltage may be approximated by the linear rise due to constant
current charging of C.
ei(0 =£ = jfct ( 7  10 )
and the time required to saturate the transistor is
h=RC^ (711)
B must be small for good linearity, and consequently we are again
forced to turn to large values of C to establish the required RC product.
To generate a 1.0msec sweep with the circuit of Example 72, C would
have to be slightly larger than 1 /if. It must be emphasized that the
RC term in Eq. (711) is not the sweep time constant but only a product
resulting from the analysis.
The approximate waveshape produced at the emitter is sketched in
Fig. 76. This point is isolated from the timing circuit and serves as a
convenient lowimpedance point from which to take the output.
Any circuit, such as the one of Fig. 74, requiring an expensive, isolated
power supply is quite unsatisfactory for general use. However, in this
particular sweep it is possible to replace the battery by a charged capaci
Sec. 72]
LINEAR TRANSISTOR VOLTAGE SWEEPS
213
tor (Ci of Fig. 77), and provided that we allow only a slight discharge
of this energy source over any cycle, the basic mode of operation will be
quite unaffected. Of course, some provision must be incorporated for
automatic recharging. Examination of Fig. 77 will show how this is
accomplished.
IWi
Its
£ 2 «r)
Fig. 76. Output voltage of the bootstrap
sweep of Fig. 74.
df
Fig. 77. Bootstrap sweep containing a
selfcharging current source.
Under the normal operating condition of this circuit, the switch
across C is closed. This permits the energystorage capacitor to charge
through R, and the conducting diode to E^,. Upon opening the switch,
the sweep begins. As the base starts rising, the emitter voltage follows
it. Since Cx remains almost fully charged over the complete sweep
cycle, the voltage at the bottom of the diode, point x, becomes
e z = En + Ae c
where the gain A is very close to unity.
Hence the very slightest increase of the sweep voltage automatically
backbiases the charging diode D x , and it may be removed from our con
sideration during the linear sweep interval. At the end of the sweep, the
switch is closed, C discharges, and the diode again conducts, finally
allowing Cx to recover toward En,.
If we neglect the small base current which is also supplied from the
charge stored in Cx, then during the sweep, charge is simply transferred
from Cx to C. Since the total charge in the circuit must remain constant,
it follows that
AQ = CEu, = Cx AE (712)
where AE is the drop in voltage across Cx over the complete sweep
interval. For best linearity AE should be as small as possible, leading
us to conclude that Ci must be very large compared with C. An adverse
effect introduced by the large storage capacity is the long time required
for circuit recovery, a time primarily determined by the time constant
ftjCi.
214
TIMING
[Chap. 7
In this sweep the mechanism employed for switching might be iden
tical with that discussed in Sec. 71. Alternatively, any of the controlled
gates would function equally well. The time constant and switching time
of the discharge path established would have to be included in any
calculation of the total recovery time of the circuit.
73. Miller Sweep. The voltage Miller sweep (Fig. 78) depends for
its proper functioning, as did the bootstrap, upon the conversion of the
current response of the transistor into proportional voltage amplification,
which is now used to multiply the feedback capacity. Before we can
(a) (b)
Fig. 78. (a) Transistor Miller sweep; (6) model holding within the active region.
discuss the overall sweepcircuit response we must evaluate the voltage
gain. The input voltage may be expressed as
ei = r n t b
(713)
where r' n = n + (1 + ff)r e . The change in collector voltage correspond
ing to this driving signal is simply
e 2 = —fliiRi
(714)
Any loading of the output by the feedback capacity has been neglected.
Solving Eqs. (713) and (714) simultaneously yields the voltage gain
A =  —r = g m R 2
Ml
(715)
Equation (715) might be interpreted as saying that the transistor
has an effective g m of /3/rij. With = 100, r e = 10, and n = 200, the
g m is 82.6 millimhos, which is so very much larger than can possibly be
obtained from any triode that a reasonable gain is ensured, even when
using very small load resistors.
After the switch is opened and the circuit becomes active, Fig. 78b
represents the complete equivalent model. In calculating the initial
base voltage by the method developed with respect to the grid voltage
Sec. 73] linear transistor voltage sweeps
of the triode sweep of Sec. 64, we obtain
£i(0+)
R* I ~lk I
215
(716)
1  A
where I is the capacitor charging current and where the approximation
is the result of substituting the high gain given by Eq. (715). The
equation indicates that there is a small positive jump due solely to the
current flow through the reflected resistance (Fig. 79a). It follows
that the reflected capacity C(l — A) will be initially uncharged.
The charging current at t = 0+ is
Eth — 2?i(0 + ) ^ Em
Rth R
7(0+) =
(717)
where £i(0+) « E Th , Rth = R \\ r' n , and E Th = r' n E hb /(R + r' n ). The
change in base voltage over the complete charging interval must be small
(b)
Fig. 79. (a) Equivalent base input circuit — Miller sweep; (fe) collector equivalent
circuit.
compared with Eti, for maximum sweep linearity. Consequently, con
stant current charging may be assumed and
Em,
«i(0
t'u En, ,
R ^ RC(1
A)
t
(718)
Since PR ^> r' u , the initialjump is quite insignificant.
From the collector model of Fig. 796, the output voltage may be
expressed as
e t (t) S En, + IRi + Aei(t)
= Etb \ 1 ~RC t )
The time required for the collector to bottom at zero is
h = RC
(719)
After bottoming, the base voltage continues rising toward a new steady
state voltage with a much faster time constant. We find these from
the model of Fig. 786 by shunting the current source /3i b with a conducting
diode. This also reduces the input resistance.
216
TIMING
[Chap. 7
When the switch is finally closed, the base immediately drops back
to zero. This same change in voltage must be coupled by C to the
collector and will drive it even further into saturation. Recovery
of the collector toward zero is quite rapid, with the time constant pri
marily depending on the resistance of the now conducting collector
base diode. Above zero, the capacitor recharges toward Em with the
time constant t* = CR2. The complete base and collector waveshapes
are sketched in Fig. 710.
74. Compound Transistors. Both voltage sweeps depended for their
linearity improvement on the gain of the feedback amplifier: the boot
strap on maintaining a gain close to unity, and the Miller sweep on
developing the largest possible negative gain. If the two appropriate
equations are examined, we conclude that for optimum operation the
t
Bit
A
~T~ 1 1 ' *
] t
^ _ }.__^__„
0 it, f ?
I 1 T 3
J J
Fig. 710. Transistor Millersweep waveshapes (ei and ei are not drawn to the same
scale).
transistor having an a closest to unity should be chosen (jS very large).
In an effort to approach unity a, and at the same time to reduce the
dependency of the circuit behavior on the individual transistor, Darling
ton proposed using a compound arrangement in place of a single tran
sistor. Figure 711 illustrates the suggested configuration: for our
convenience in the later discussion the approximate current flow into and
out of each transistor element is indicated. The arrows mark the actual
direction of the current in the pnp transistors shown.
The small base current of the primary transistor (Ti), i e i(l ~~ «i),
is amplified by the correction transistor (T2) and added to the output of
T,
«2(l«l)»,.l
(lor 2 )(lori)t c i
6
Sec. 74] linear transistor voltage sweeps 217
T\. Thus the composite collector current is composed of two terms :
ie = id + id = «i»«i + (1 — ai)ati.i
The first term is the normal current
transmission through any transistor, m y, *i>.i
and the second term represents a e °
small additional correction current
flow from T2 that raises the overall „ „ v "j~
output to a value much closer to the  ,
input driving current. We should A A
note that as long as the individual V__y
transistor a's are less than 1, the
total output current will always be less
than i,i. The two transistors, taken _ _ „ _
. ii i * IG  711. Compound transistor cir
together, can be said to act as a single cu i t
compound transistor, one having
<*c = <*1 + (1 — «l)«2
= 1  (1  ai)(l  a,) (720)
For example, when a\ = 0.98 and a 2 = 0.97, the composite current gain
is 0.9994.
Additional transistors may also be incorporated for further correc
tion, each amplifying the base current of the previous transistor and
adding it to the overall output. They raise a c by multiplying the second
term of Eq. (720) by additional factors of the form (1 — a,). But regard
less of the number used, a c will never quite reach unity. These addi
tional transistors have a progressively decreasing effect on a c , and there
fore the composite unit is usually composed of no more than two or three
junction transistors.
Stability of the composite element with respect to the individual a's
is also much better than that of a single transistor. If only small varia
tions are considered, then
5oT  1 ~ " 2 sr, = 1 ~ a > (7 " 21)
Using the figures given above, a 1.0 per cent change in ai would cause
only a 0.03 per cent change in the overall current gain a e . Equation
(721) is not valid with respect to large variations of ai or a 2 , and in this
case the effect on a e would have to be found by evaluating Eq. (720) over
the expected range of ai and/or a^.
The composite transistor is not an unmixed blessing; further examina
tion of the circuit of Fig. 71 exposes serious drawbacks which severely
limit the possible applications. For example, the temperaturedependent
218 timing [Chap. 7
reverse collector current /,o of T% still flows unchanged at the base of the
composite unit. Since the value of ft is very large, only a very small
base current is needed to control the complete collector current flow.
But lea may well be of the same order of magnitude, making the problem
of temperature stabilization extremely difficult.
Furthermore, it can be shown that the input impedance at the com
posite base may be approximated by
r in .„ S [r.i(l + ft) + r bl ](l + ft) (722)
As ft and ft are very large, the base input will no longer approximate
the ideal short circuit which is desirable in a currentcontrolled device.
But for just this reason, this circuit configuration is more convenient
for use as a voltage amplifier; the loading of the external source decreases
with the increase in ft (refer to the requirements given in Sees. 72 and
73). If ever a higher impedance were needed, additional padding
resistance would be inserted in series with the emitter. It would be
multiplied by the product (1 + ft)(l + ft) as it is reflected into the
base circuit.
PROBLEMS
71. Compare the sweep linearity if the transistor sweep of Fig. 71 is first used in a
grounded base connection and then as a grounded emitter circuit. In both cases R$ is
adjusted to make the sweep period 1.0 msec. The other circuit parameters are a =
0.98, r c = 0.5 megohm, C = 0.1 4, E bb = 10 volts, and E cc = 10 volts. Specify
the required values for Ri.
72. (a) Plot the locus of operation (i c versus e„) of the sweep of Fig. 72 on the
collector characteristics and evaluate the sweep and recovery times. The component
values are ai = at = 0.99, r c = 1 megohm, C = 1 nf, R = 2 K, R 2 = 5 K, Ebb =
10 volts, and E cc = 10 volts.
(6) Modify this circuit so that a very short input pulse will periodically discharge
the capacitor, with the sweep starting immediately upon the termination of the pulse.
Sketch the new circuit. Specify the necessary value of Ri if the discharge must be
complete within 10 /isec.
(c) Compare the necessary power requirements for the circuits in parts o and 6.
Which one offers the most efficient operation if the spacing between sweeps is O.lti,
ti, 5ti? (ti is the sweep period.)
73. The switching transistor in Prob. 72o presents a resistance of 25 ohms from
the collector to the emitter when saturated.
(o) How will this term affect the output waveshape and period? Write an expres
sion for the sweep interval in general terms, calling the conduction resistance r„ and
then evaluate this equation.
(6) Plot both the output waveshape and the locus of operation for the above circuit.
74. We wish to evaluate the performance of the sweep of Fig. 712 to see whether
it yields the same response as the circuit of Fig. 71 without having recourse to two
power supplies. We have at our disposal a transistor having p = 50 and r„ =
1 megohm, a 20volt power supply, and a capacitor of 1,000 wit.
LINEAR TRANSISTOR VOLTAGE SWEEPS
219
(a) What values of R. and Rb are needed to give a sweep amplitude of 15 volts and
a sweep duration of 15 Msec?
(6) What value of L is required so that the change in emitter current over the com
plete sweep period will be less than 0. 1 per cent?
(c) Sketch e%, specifying all values.
?+20v
Fig. 712
76. The bootstrap sweep of Fig. 74 has the following component values:
n  200
E = 6 volts
r„ = 1 megohm
E bb = 12 volts
r, = 20
R. = 1,000
= 50
C = 0.04 4
R = 800
(a) Sketch the sweep output, labeling all voltages and times.
(6) Evaluate the sweep nonlinearity.
76. (a) Express the sweep nonlinearity of the circuit of Fig. 74o as a function
of (3.
(6) The circuit and parameters are En = E = 10 volts, »■'„ = 1,000, R, = 1,000,
and R = 500ohms. Forwhat range of /Swill the NL fall between 0.5 and l.Opercent?
77. Prove that the bootstrap sweep of Fig. 74 may be represented by the model
of Fig. 713. Find the values of R n , A, and / for the circuit of Example 72. Com
pare the sweep duration and nonlinearity with that given in the text.
Fig. 713
78. (a) In the bootstrap sweep of Fig. 74, the bias source E is equal in value to
Ebb, but it remains constant, whereas Ebb varies by + 20 per cent from its nominal "
value of 10 volts. Calculate the change in sweep duration when r u «• 1,200, «■ 50,
R = 1 K, R, = 1 K, and C = 0.05 4.
(b) The sweep of part a is connected in the configuration of Fig. 77 with C\ =
10 /if Compare the sweep duration at the limits of Em with the answers found in
part a.
220
TIMING
[Chap. 7
79. Figure 714 represents an alternative mode of switching the bootstrap sweep
of Sec. 72. However, once the collector is opened, the conducting baseemitter
diode presents a resistance of only 20 ohms, instead of the 1,000 ohms (r n ) seen with
the switch closed. Plot the emitter voltage and the controlling current i'i to scale,
after the switch is closed at t = 0. Pay particular attention to the circuit behavior
as the switch closes. The other components are R = 500, R, = 2 K, /S = 100,
C = 2 id, and E = E bb = 20 volts.
Fig. 714
710. The bootstrap circuit of Fig. 77 uses a transistor with the following parame
ters: r' u = 900, /S = 10. With this transistor and with R, = 5 K, R  1.5 K, C =
1 id, and Ey, = 20 volts, find
(a) The waveform at the emitter when the switch S is opened at t = and closed
when the voltage is equal to E bb . (Note that is relatively small, which invalidates
some of the approximations made in Sec. 72.) Label all time constants and find the
time when S is closed.
(ft) The sweep nonlinearity.
711. In the Miller sweep of Fig. 78 the switch is opened at t = and closed once
the collector falls to 1 volt. The transistor employed has the identical characteristics
of the one used in Prob. 75, and the other circuit values are Ri = 10 K, R = 5 K,
C = 0.01 id, and E bb = 15 volts.
(a) Sketch the collector waveshape, making all approximations given in Sec. 73.
Calculate the sweep nonlinearity.
(b) Repeat part a if R is reduced to 1,000 ohms and C increased so that the period
remains the same as in part o.
(c) If r, varies by a factor of 2 over the dynamic range of the transistor, how will
this affect the sweep period? Give a qualitative answer.
h20v
r.lO
r 4 100
10 K£ ,„ f £10K 050
=lv
Fig. 715
712. We shall use the circuit of Fig. 715 as an alternative to the Miller sweep of
Fig. 78. The external emitter padding resistance minimizes the effects of the chang
LINEAB TRAN6ISTOB VOLTAGE SWEEPS 221
ing n and r« during the collector rundown. Furthermore, to prevent the transistor
from being driven completely into saturation, a bottoming diode is connected to the
collector. For the circuit values given below, calculate the complete collector and
base waveshapes, from t = until steady state is reached.
713. Repeat Prob. 710 when the single transistor is replaced by the compound cir
cuit of Fig. 711. Each transistor has the identical parameters given in Prob. 710.
Do the approximations of Sec. 72 now hold? How would a third transistor affect
the sweep period? Discuss the effects of the increasing input impedance on the
sweep waveshape.
714. Prove that the maximum base input resistance of a composite transistor,
composed of two identical units, must be less than the collector resistance r« of one.
Show that this holds regardless of the size of the padding resistance inserted in series
with the composite emitter.
716. The composite transistor of Fig. 711 is composed of a highpower transistor
Ti with its characteristics improved through the addition of T 2 . If these two tran
sistors have the parameters listed below, what is the equivalent base input impedance,
emitter input impedance, basetocollector current gain, and maximum power dissipa
tion of the composite unit?
a,  0.95 a: = 0.99
n = 100 n = 400
r. = 20 r. = 15
P m  5 watts P m = 200 mw
716. Calculate the composite characteristics of the npn and pnp transistor
shown in Fig. 716 (ri„,„ n»,i>, «„, and e ). Assume that the two units are identical
in all respects except the direction of current flow. Would this configuration be help
ful in linearizing any of the voltage or current sweeps discussed in this chapter?
Explain how.
BIBLIOGRAPHY
Clarke, K. K., and M. V. Joyce: "Transistor Circuit Analysis," AddisonWesley
Publishing Company, Reading, Mass., in press.
Darlington, S.: Patent No. 2,663,806 (assigned to Bell Telephone Laboratories).
Nambiar, K. P. P., and A. R. Boothroyd: Junction Transistor Bootstrap Linear
Sweep Circuits, Proc. IEE (London), pt. B, vol. 104, pp. 293306, 1957.
CHAPTER 8
LINEAR CURRENT SWEEPS
The electric deflection of a highenergy electron beam imposes a
severe strain on the circuit designer in that it requires an excessively
large sweep amplitude. For this reason magnetic deflection, which is
produced by a linearly increasing current in the deflection coil, is used
instead. The most common example is a television receiver where the
raster is generated by the magnetic deflection of the beam: a single
horizontal scan taking about 53 /isec and a vertical scan about 16 msec.
Radar display scopes and some electromechanical systems employ
similar deflection circuits. Furthermore, a coil excited with a linear
current may be mechanically rotated, thus generating a spiral sweep.
Our treatment of current sweeps in this chapter will closely parallel
the development of the various voltage sweeps, with, however, an
inductance replacing the capacitor as the basic timing element. We
should expect our thinking to be influenced by the previous discussion
and should feel free to adapt any existing circuit to our current need.
For purposes of comparison and in order to gain perspective, the reader
might occasionally refer back to the appropriate sections of Chaps. 6
and 7.
81. Basic Current Sweeps. If the ideal inductance did exist, as does
the almost ideal capacitor, then the problem of producing a linearly
increasing current would become trivial; simply switching a constant
voltage across the coil would suffice. Because E = L di/dt, the current
would immediately become
i = \t (81)
Unfortunately all coils have distributed winding resistance and inter
winding and stray capacity; a typical ironcore deflection coil of 50 mh
may have a resistance of 70 ohms and an effective capacity of 200 ft/if
These parasitic elements are generally represented by the lumped param
eters (Ri, C) shown in the basic sweep circuit of Fig. 8la, and their effects
on the sweep waveshape must be accounted for in any analysis.
Since we are interested in a linear sweep, the portion of the cycle
devoted to the initial charging of C must be much smaller than the
222
Sec. 81] linear current sweeps 223
time required for the coil current to build up to its final value. Immedi
ately upon closing the switch in Fig. 8la, the complete circuit current
flows into C. Thus the beginning of the sweep may be represented by
the model of Fig. 816, which implicitly assumes that the start of the
£.=.
(b)
R,
E—
4 \ \
e L gi
(c) (d)
Fig. 81. (a) Basic current sweep; (6) approximate model holding during initial
charging; (c) model representing the currentsweep region; (d) recoveryregion model.
inductivecurrent buildup will be delayed until the terminal voltage
reaches E. The time constant of this circuit,
ti = R.C
(82)
is quite small, and the delay is proportional (h — 4ti) ; even when charg
ing through 500 ohms, the 200niti capacity introduces a delay of less
than 0.5 /usee (Fig. 82).
Sweep Region. We can now assume that the circuit enters into its
sweep region, which is defined by the model given in Fig. 8lc. Neglect
ing the small current change occurring during the charging of C, the
time response becomes
it
E
R, + Rl
(1  e"") tj =
tli ~\~ R»
(83)
To justify our separating the response into these two segments,
the sweep time constant ti must be much longer than r\. In the case
considered above (L = 50 mh, R L — 70 ohms, and R, = 500 ohms),
T2 = 88 usee. If R, is further reduced, the separation between n and r 2
increases and the assumption made above becomes even more valid.
224 timing [Chap. 8
Moreover, in order to ensure good sweep linearity, the charging interval
U must be limited to some small fraction of t*.
Suppose that we employ this coil in a deflection circuit where the
current must reach a peak of 200 ma in 53 Msec. From Eqs. (83), the
necessary supply voltage is found to be
E
570I m
= 252 volts
The inductive component of the voltage decays to 138 volts over the
sweep interval. An additional 14 volts (200 ma through 70 ohms)
appears across the resistive component, raising the final coil voltage to
152 volts.
Fig. 82. Sweep current and voltage waveshapes for the basic sweep of Fig. 81 (switch
in position 1).
The waveshapes produced during this portion of the cycle are sketched
in Fig. 82. Note the delay in starting the sweep and the compara
tively small voltage change across the complete coil and the large voltage
change across the inductive component.
Sweep Recovery. At the end of the required sweep interval, the switch
is thrown to position 2. The response of the coil with its parasitic
capacity and external damping, Rd, is found from the single node equa
tion of the circuit in Fig. 8ld. We assume, in writing Eq. (84), that the
series coil resistance will have but little influence during the portion
of the recovery time with which we shall be concerned.
% + h + ir = C
+ if edt + w D = °
(84)
The roots of Eq. (84) are
Pi, 2 =
1
2RdO
± V(^c) 2  ic (8 " 5)
Two bounding possibilities serve to delineate the response character
istics of the network. In one case. (l/2.RpC) 2 < 1/LC and the circuit
Sec. 81] linear current sweeps 225
exhibits a damped sinusoidal response. Here the poles are located at
Pi,2 = — a + jfi
1
Where " = 2R D C f> = Vo>S«* »<? = LC
The complete solution of Eq. (84) may be expressed as
4 = Ae<" cos (fit + <t>)
If the system is but slightly damped, a <K /S. By including the initial
conditions, which are the final voltage and current reached during the
linear sweep {E f and 7 m of Fig. 82),
ih = I<n.e~ at cos wo< (86)
e L ££  Jjj I m e"> sin wot (87)
Since the damping factor a is so very small, the maximum coil voltage
occurs when sin u4 — 1 (in 34 cycle of / or at t = w y/LC/2) :
eL.„„ ^  Jg In, (88)
For the coil considered above, this peak reaches almost —3,200 volts
only 4.98 jusec after recovery begins. Figure 83a illustrates the rapid
oscillation and slow decay of the recovery portion of the sweep waveshape.
A second bounding location of the roots of Eq. (84) occurs when
Rd is adjusted to critically damp the response (Rd = \/£/4C). The two
roots are now identical, both lying on the real axis at
1
Pl.2 = —
\/LC
When the initial conditions are included, the solutions for the critically
damped case become
%L
ei =
\L + VLC)
(JL. + UY
exp VW (8 " 9a)
exp^ (S%)
The peak voltage is much smaller, and it occurs sooner; for the same coil
considered above, it reaches —2,380 volts at t = 3.16 Msec. Figure 83b
shows this portion of the complete waveshape. As Rd is further reduced,
the poles separate along the real axis. Even though the peak voltage is
226
TIMING
[Chap. 8
reduced below the value shown in Fig. 836, the increase in the time
required for final recovery generally prevents the use of a very small
damping resistance.
In comparing Fig. 83o and b, we conclude that the fastest possible
recovery corresponds to a single half cycle in the completely undamped
case. For periodic sweeps the "switch" is returned to position 1 at the
time that the coil current reaches its minimum point and the capacitor
voltage returns to zero. The current will build up from approximately
— /„ instead of from zero, and the sweep amplitudes and time will
increase accordingly.
3,200 v
T"*\J
E r
\ 3.16 /isec t
2,380 V
I
(a)
(b)
Fig. 83. (a) Recovery waveshape when lightly damped; (6) recovery waveshape
when critically damped.
82. Switched Current Sweeps. Figure 84 illustrates the application
of a transistor to the basic sweep of Fig. 81. We shall first assume that
the circuit is initially inert; i.e., it contains no stored energy. When a
pulse is injected, the transistor switches from cutoff to saturation, con
necting the full supply voltage across the inductive component of the
circuit. (See the model of Fig. 85a.) Because the saturation resistance
of the transistor r, is quite small, the delay in starting the sweep, due to
the stray coil capacity, will be insignificant. Therefore, except during
the oscillatory recovery interval, our discussion of this circuit will com
pletely neglect C.
Immediately after switching, the operating point moves to the origin
of the collector voltampere characteristics and the current starts increas
ing toward the intersection of the steadystate load line, —1/Rl, with
the saturation resistance line of the transistor characteristics (Fig. 86).
Sec. 82] linear current sweeps 227
This steadystate current and the charging time constant may be found
directly from the model of Fig. 85a.
•/«»i —
Em
Rl + r,
T\
Rl + t,
(810)
In general, Rl 3> r, and both terms of Eqs. (810) are primarily deter
mined by the coil and battery; they are almost completely independent
of the transistor.
L,R L ,C S
Fig. 84. A switched current sweep — the transistor is driven from cutoff to saturation.
fSoooooj WV
=E u
=rE u
(a)
r — nmum^
(b)
WA
T
(c)
Fig. 85. Models for the sweep of Fig. 84. (a) Charging region — transistor saturated ;
(6) recovery region — transistor cut off; (c) recovery region — collectorbase diode
conducting.
The collector current buildup is permitted to continue until it reaches
I m , a value somewhat less than ali, where Zi'is the peak injected emitter
current. If the current buildup continues past this point, the transistor
will enter its active region and the increase of the collector resistance
(from r s to r c ) will radically distort the sweep waveshape.
After the input pulse is removed, the transistor becomes backbiased.
Since the only damping present is the very large reverse resistance of the
transistor and the small coil resistance (Fig. 856) , the output waveshape
is a damped oscillation similar to that shown in Fig. 83a.
228
TIMING
[Chap. 8
In this region the coil current and collector voltage follow the elliptical
trajectory drawn on the collector characteristics of Fig. 86. At point A,
the energy of the complete system,
w L = y 2 Li n
(811)
is stored in the magnetic field of the coil. Along the path AB the stored
energy is transferred to the electric field of the capacitor. As the coil
Fig. 86. Trajectory of the collector voltage and coil current for the switched sweep
of Fig. 84. The dashed portion is the path taken in the oscillatory recovery region;
the solidline segments, the path followed in the linearsweep region.
voltage reverses and builds up to the negative peak of E v , the coil current
decreases to zero. At point B,
W c = }iCE p *
(812)
From Eqs. (811) and (812),
E.
which is identical with the result given in Eq. (88).
Because of the transistor's limitations, only a small voltage can be
tolerated during retrace. By shunting the coil with additional resistance
and capacity the peak can be reduced to an acceptable value, but this
also reduces the frequency of oscillation and increases the portion of the
sweep which must be allocated to recovery. If this time must be mini
mized, while still limiting the peak voltage, either smaller inductance
Sec. 82] linear cukbent sweeps 229
deflection coils can be used or two transistors may be placed in series,
reducing the drop across each to onehalf the total voltage present.
Along the second portion of the elliptic trajectory BD, the energy is
retransferred from C to the magnetic field of the deflection coil. The
voltage now decreases, and the current builds up to — I m . The dashed
portion is the path taken in the oscillatory recovery region and the solid
line segments, the path in the linearsweep region.
Slightly after onehalf cycle of the oscillation (Figs. 86 and 87),
the positive voltage developed across the coil exceeds the negative
supply and the collector becomes forwardbiased with respect to the base.
A conduction path through this diode now exists irrespective of the
emitter condition. The model holding changes from that of Fig. 856
to the one of Fig. 85c. Current now builds up from the negative peak of
— I m toward a new steadystate value
J««2 =
with a new time constant
Ti =
Eu
RL + Tf
R L + r,
The charging path is now along the straightline segment DO in Fig. 86.
In order for the two portions of the sweep to have an equal slope and the
same steadystate current, a transistor with r/ = r, must be selected.
When R L is large, it will predominate and a much greater degree of
unbalance in r/ and r, can be tolerated.
To generate a periodic sweep, the emitter must be periodically switched
from off to on. The time of the application of the input gating pulse
is not extremely critical provided that it occurs before the coil current
rises to zero. Under these conditions the transistor changes state
smoothly, from its operation as a forwardconducting collectorbase
diode to its saturation region. Buildup continues along the path DO A
to I m . Along this line segment the charging equation is
hit) = I..  (/.. + I m )e» (813)
And from Eq. (813), the time required for the current sweep to build up
from — J„ to I m is
<i  t In {" + { M (814)
At the end of the linearsweep interval the excitation is removed
and the circuit recovers as discussed above. The portion of the period
230
TIMING
[Chap. 8
devoted to the flyback constitutes only one half cycle of the oscillatory
wave, or
1 IT
U =
2/o
«0
= tVlc
(815)
We have seen how a periodic excitation leads directly to a periodic
sweep with half of the linearsweep interval depending on the saturated
transistor (external excitation) and the other half upon the selfexcitation
(stored energy) of the diode characteristics.
Fig. 87. Periodic sweep produced by the circuit of Fig. 84 upon squarewave exci
tation.
We should note that this sweep circuit is extremely efficient. During
the half cycle that the transistor is saturated, energy is transferred from
En to the coil. During the half cycle when the reverse current flows
through the conducting collectorbase diode, the stored energy is returned
to the power supply. Of course, some power is dissipated over each
half cycle in the circuit resistance.
Satisfactory switching may be obtained from a square wave of current
coupled into the emitter through an emitter follower. The sweep pro
duced is shown in Fig. 87.
Example 81. We wish to see whether the sweep of Fig. 84 can be used in the
horizontaldeflection circuit in a portable television receiver. Of the 63jisec total
period, the linear sweep requires 55 /isec, with 8 jisec allowed for the sweep recovery.
Sec. 82] linear current sweeps 231
The deflection yoke, which must be excited by a peaktopeak current of 1.0 amp has
the following parameters: L = 0.5 mh, R = 4.5 ohms, and C, — 50 pid.
(a) With this coil, what supply voltage must be used?
(b) How can the peak reverse voltage be minimized?
(c) What is the sweep nonlinearity?
Solution. Assume that both the saturation and the forward resistance of the
transistor employed are 1 ohm. The time constant of the sweep interval is n =
91 /jsec. Solving Eq. (814) with I m — 0.5 amp yields
/„ = 1.7 amp
Hence Eu, = 1.7(r, + Rl) = 9.35 volts.
As C is increased, the peak reverse voltage decreases. The upper limit of C is that
value that increases the half period of the recovery sinusoid to 8 /usee. From Eq.
(815),
c »=z6) 2=0013 ^
Thus the peak voltage becomes
^ =  5 VrHTp = 106volts
Sweep nonlinearity is given by Eq. (520) :
NL = 12.5% X 5 %i = 7.5%
For acceptable linearity the coil inductance must be increased or the totel resistance
decreased.
Pentode Switched Sweeps. If a pentode is used as a switch (Fig. 88),
because of the higher saturation resistance the start of the sweep will be
slightly delayed. As the stray capacity charges, the operating point
moves to the origin of the voltampere characteristic and the full supply
voltage appears across the coil. This effectively places the tube in the
region where it has a small value of r p , i.e., along the 2?&, miI , line, where
r p drops to r' p . When a power pentode is used as the switching tube,
r' p may be as low as 100 or 200 ohms. (The process described is similar
to the bottoming of the pentode Miller sweep of Sec. 67.)
The coil current increases exponentially from zero, with the time
constant r = L/(r p + Rl), and as it does so, the operating point travels
up the Et. miB line of the pentode characteristics toward the intersection
with the load line. In order to maintain an almost linear buildup, the
sweep will be terminated long before it reaches this steadystate current
and before the tube becomes active.
The removal of the controlgrid signal turns the tube off. The large
pulse now developed at the platecoil current can be limited to a safe
value by the addition of a suitable damping resistance. In the circuit
of Fig. 88a, it is included in series with a diode, which is connected
so that it will conduct only when the coil voltage reverses.
232 timing [Chap. 8
Since the tube cannot tolerate any reverse current flow, if we wish
to adapt the pentode circuit for the most efficient recurrent operation,
the coil must be shunted by a path which will support the sweep when
%l < 0. In the transistor circuit of Fig. 84 this path was established
L^^
(a) (b)
Fiq. 88. Pentode switched sweeps, (a) Singleshot sweep circuit showing the con
nection of the damping diode; (6) periodic sweep showing the connection of the
energyrecovery diode.
through t9e collectorbase diode; by analogy, an energyrecovery diode
can be connected across the tube (Fig. 886). Just as in the transistor
sweep, the diode will start to conduct slightly after the first half cycle
of the damped sinusoid. At this time the coil voltage exceeds Em,, mak
ing the plate of the pentode negative with respect to ground. A com
parison of the vacuumtube circuit of Fig. 88
with the transistor sweep of Fig. 84 convinces
us that the transistor is much more suitable
for periodic current sweeps than the vacuum
tube. The only advantage offered by the tube
is that it can tolerate large reverse voltages
during recovery.
83. Current Sweep Linearization. The
constant voltage charging of Sec. 81 is not a
very satisfactory sweep for critical applications
because of its delay in starting and its inherent
nonlinearity. We now propose to reexamine
the basic sweep of Fig. 8106 and ask, What applied wave shape will
ensure a linearly increasing current flow?
In order to minimize the sweep delay, the initial portion of the excita
tion waveshape should be a voltage impulse. The impulse, shown in
Fig. 89, is a pulse whose amplitude increases toward infinity as its
duration decreases toward zero. Its area remains constant during the
S(t)
AT
AT
Fig. 89. Defining the unit
impulse.
Sec. 83]
LINEAR CURRENT SWEEPS
233
limiting process. It follows that the unit impulse S(t) may be defined
by the area integral
CO* fAT 1
jo W di =j AT dt = l
(816)
where, in the limit, AT = 0+  0.
If we excite the coil through the source impedance R, with a voltage
impulse of area K, then the stray capacity would be charged by the
current
. _ KS(t)
% ~ R.
From Eq. (816), the voltage across C is given by
In order for this voltage to equal E at t = + , the weight of the impulse
must be
K = CR.E
It is impossible to generate the ideal impulse, but it is possible to inject
a finiteamplitude pulse at the institution of the sweep to speed the
charging of C. Any largeamplitude pulse will reduce the sweep starting
time by an appreciable factor.
Impulse
Wm
(a)
(b)
Fig. 810. (a) Required drive for the ideal current sweep; (ft) equivalent circuit of the
sweep.
To find the required drive voltage over the sweep region, we assume
that the desired linearly increasing current is already flowing.
i = kt = =? t
fi
(817)
Substitution of this linear term into the circuit loop equation results in
e,„ = L% + Rii = Lk + RJct
at
(818)
234
TIMING
IChap. 8
where Ri = R L + R,. Equation (818) defines the trapezoidal driving
voltage shown in Fig. 8 10a. The first term in the above equation
{Lk = E) is the magnitude of the pedestal; i.e., it is the constant voltage
which must be developed across an ideal inductance if the current is to
increase linearly with time. The second term represents the linearly
increasing voltage drop appearing across the resistive components of the
circuit due to the current sweep produced in the coil.
The individual components of the trapezoidal drive voltage necessary
for a linear sweep may be generated separately, added together, and
used to excite the coil. For example, by integrating a rectangular
voltage wave in the input circuit of the emitter follower of Fig. 811,
or alternatively in the grid circuit of a cathode follower, the voltage
impressed across the coil would be of the proper trapezoidal form. The
Fig. 811. Emitterfollower drive circuit for a current sweep.
low source impedance of the driver stage limits the peak excursion of the
output voltage to a reasonable value. A starting impulse must be
separately injected. Many other modes of signal generation and coupling
are also possible — through a transformer from a highimpedance source,
capacitive coupling to eliminate the dc component in the coil, or directly
from a pushpull circuit.
Example 82. The emitter follower used in the sweep of Fig. 811 employs as its
active element the composite transistor of Sec. 74. Each unit has an a = 0.98.
The 200mh 30ohm deflection coil must be excited by a linearsweep current which
reaches a maximum of 0.5 amp in 20 msec. In generating the input signal, the end
of JBi is connected directly to the 50volt supply. What must Ri, R s , and C be to
satisfy the necessary sweep conditions?
Solution. With the composite transistor, the large value of a e (0.9996) given by
Eq. (720) means that the overall gain is extremely close to unity. Consequently,
the reflected input impedance of the emitter load (Rl and L) back into the base will
be large enough so that its loading of the input RC network will be negligible. From
Eq. (818), the height of the pedestal must be
E  Lk  200 X 10« X
0.5
20 X 10" a
5 volts
Sec. 84] linear current sweeps 235
The maximum height of the triangle, neglecting the small emitter resistance, is
E, m = R t I m  30 X 0.5 = 15 volts
Assume that the initial current through the RC network is 1.0 ma. To obtain a
5voIt jump at the base input upon excitation,
D 5 volts _ „
1 ma
The remaining 45 volts must be developed across the other input resistor :
= 45yolts _ 45 K
1 ma
The base must charge from 5 to 20 volts in the 20msec sweep duration. From the
exponential charging equation,
,5020 .„ .
f i — t In gf — = ■" 49.5 msec
5U — o
Since Ri + R t = 50 K, C = 0.99 jif. We should note that the sweep duration is
independent of the 6.7msec coil time constant; in fact, the approximately linear
sweep is now three times as long.
84. A Transistor Bootstrap Sweep. For a second method of linear
izing the current sweep we might consider a circuit which is the dual
of the RC voltage sweep of Fig. 62, i.e., the parallel combination of a
coil and a conductance charged from a constant current source (Fig. 812).
This circuit's node equation may be
written as
e £ =^=^ (819) 7 tQ
Ml f
At this point Eq. (819) should be
j •j.i. i.\. i .,, Fig. 812. Simple LG current sweep,
compared with the equation written
for the voltage sweep [Eq. (66)] and the argument employed with respect
to that circuit reread.
We know that bootstrapping generates an extremely linear voltage
sweep. Since we now have at our disposal a currentcontrolled active
circuit element, the transistor, we should consider the possibility of
developing the dual circuit, a current bootstrap, as our current sweep.
Choosing Eq. (819) for the starting point, we see that the replacement
of /' by / + A c i L provides the correction term which is so essential.
When this is done,
eL = f + A * L  h (820)
where A c is the current gain of the feedback amplifier. Figure 813a and
b illustrate the current bootstrapping process defined in Eq. (820). It
should be noted that the role of the current amplifier is analogous to that
236
TIMING
[Chap. 8
performed by the voltage amplifier in the voltage bootstrap. Both sup
ply the additional energy made necessary by the dissipative elements of
the external circuit. By this means the system maintains either constant
current or constant voltage charging of the energystorage element.
Since under these circumstances the dissipative energy remains constant
(PR or EH3), the system must supply an everincreasing amount of
energy for storage in the capacitor or inductance. Unless an active cir
cuit element is present to cancel the effects of the circuit resistance (which
limits the available storage energy), it is impossible to achieve an abso
lutely linear sweep.
l+A c i L
G  Qt /+A ^
EIR
(a) (b) (c)
Fig. 813. Basic current bootstrap circuit, (a) Circuit connection; (6) schematic
representation ; (c) TheVenin equivalent circuit.
The role of the amplifier becomes somewhat clearer if we solve for the
Thevenin equivalent of the current generator and its internal conductance
in Fig. 8136. One of the two voltage generators resulting is of the form
e„ = RA c tL
where R = 1/G. But since this voltage rise is proportional to the series
current flow, it may be replaced by a negative resistance — RA C , as
shown in Fig. 813c. The amplifier's action serves to reduce the overall
series resistance and to increase the circuit time constant. The current
sweep is now defined by
il =
E
Rl + R(l  Ac)
(1  e«")
R L + R(l  A e )
It is immediately apparent that there will be no improvement over the
first circuit discussed unless A c is positive and greater than unity. When
A c < 1 there will actually be deterioration of the sweep quality as a
result of the decrease in the steadystate current and the time constant.
Since both terms change in the same direction, the slope at the origin
remains invariant with respect to any instability of A c . Any improve
ment in the sweep permits a larger amplitude output before the nonlinear
ity becomes excessive.
Sec. 84] linear current sweeps 237
Ideal constant voltage charging occurs when there is complete can
cellation of the resistance. From Fig. 813c we see that the necessary
amplifier gain is
A c = 1 + ^ (821)
Usually R 3> Rl, and consequently the current gain required for linear
charging will be but slightly greater than unity. Since we are dependent
on the cancellation of two terms of equal magnitude for sweep lineariza
tion, any instability in i„ will have pronounced repercussions on the
current waveshape, particularly on the time required to reach a predeter
mined final value [Eq. (610)]. Generally, the amplifier is adjusted so
that the net resistance will be small but positive and so that any expected
gain instability will never result in a net negative circuit resistance. The
high degree of stabilization necessary, if the sweep time is to be very
long, sets extremely stringent requirements on the base current amplifier.
Previously, we proved that a trapezoidal voltage drive must appear
across the physical inductance when a linearsweep current flows through
it [Eq. (818)]. Substituting the assumed ideal current u, = kt into the
bootstrap equation (820) and equating the result to the known drive
signal affords another method of determining the essential system
conditions:
I + (A c  l)kt
«l = RiJd + Lk =
G
Comparing similar terms on both sides of the above equation yields two
relationships :
A e  1 = R L G = ^ (822)
I =LkG (823)
The first of the above equations establishes the necessary system gain,
and, as expected, it is identical with Eq. (821). The second condition
[Eq. (823)] expresses the constant voltage charging of the inductive
component of the external circuit. By rewriting this in the form
Hi
E = IR = L%= Lk
at
it also offers a convenient design criterion for the choice of the parallel
resistance.
The positive current gain of a single transistor is necessarily limited
to somewhat less than a, and consequently a multiplestage amplifier
must be used in this bootstrap sweep. Figure 814 illustrates an appro
priate circuit, which employs current feedback both to increase the gain
238
TIMING
[Chap. 8
stability and as a means of adjusting the gain to the optimum value.
This will generally be somewhat less than that given by Eq. (821),
thus allowing a margin for any remaining amplifier instability. Since the
coil resistance will not be completely canceled, if originally Bl was small,
little or nothing will be gained by bootstrapping and the switched sweep
of Fig. 84 might prove to be just
as satisfactory.
The current sweep discussed
above is the dual of the bootstrap
voltage sweep treated in Chaps. 6
and 7. By analogy it would seem
likely that a currentcontrolled
Miller sweep, depending for its
timing on the inductance, might
also be practical. However, the
impossibility of isolating the coil
resistance causes any operation on
Fig. 814. Base amplifier for a current
bootstrap circuit.
the inductance to be accompanied by an equal effect on the associated coil
resistance. Multiplying or dividing both L and R L by the same factor
leaves the time constant unchanged. Because there is no improvement,
this method of linearization is never used. It is worth noting that not
all forms of voltageoperated systems can, in fact, be converted to
their duals. Often, as in this case, the unavailability of the needed ideal
circuit element (pure inductance) will be the decisive factor.
Fig. 815. Current sweep generated from a constantcurrent source.
85. Constantcurrentsource Current Sweep. A contrasting line of
thought actually obviates the generation of a current sweep. The tube
or transistor is utilized solely as a voltagetocurrent converter. A highly
accurate voltage sweep is generated elsewhere and applied as a drive:
in Fig. 815 the constantcurrent output of the transistor furnishes a
driving current proportional to this voltage. We thus bypass the need
to develop separately the trapezoidal voltage across the coil, the negative
resistance in series with the coil, or any other correction term necessi
tated by the nonideal inductance. Moreover, by restricting the sweep
generation to voltage waveshapes, we are able to employ an almost ideal
Sec. 85] linear current sweeps 239
energystorage element (i.e., a capacitor), with consequent ease of
linearization.
The current flow through L, found from the model holding for the
active region, is
R ~ (824)
where E d is the drop across the inductance. In general, Ri <5C r c and
the small current component due to Eu, will be negligible; the coil current
becomes a linear function of the driving voltage.
At the end of the sweep cycle the input voltage drops to zero. The
turnoff process is identical with the removal of the excitation in the
switched sweeps of Sec. 82. Here also the recovery would be controlled
either by an external damping diode and resistance or by a recovery
diode for recurrent sweeps.
PROBLEMS
81. A coil having an inductance of 100 mh, a resistance of 20 ohms, and stray
capacity of 200 nid is charged through a source impedance of 1,000 ohms (Fig. 81).
The sweep current must reach 150 ma in 50 fisee.
(a) Specify the necessary charging voltage and calculate the sweep linearity.
(6) Sketch the current and voltage waveshapes when the circuit is critically
damped.
(c) Repeat part 6 when the damping resistance is reduced to 1,000 ohms. Specify
the peak voltage and the time at which it occurs. Make all reasonable approxima
tions. (Hint: Treat in the same way as the doubleenergy circuit of Chap. 1.)
82. The coil of Prob. 81 is shunted with additional capacity so that one half cycle
of the oscillatory recovery lasts for 60 iisec (fi d = «). Calculate the peak voltage
and compare with the value found for the coil without additional shunting capacity.
What does the additional capacity do to the initial sweep delay?
83. The transistor used in the sweep of Fig. 84 is excited by a 1.0msecperiod cur
rent square wave. Both r. and r, at the collector are 5 ohms, r„ = 100 K, a = 0.97,
Eu, = 20 volts, and the coil parameters are L = 0.05 henry, R = 10 ohms, and C =
500 wf.
(a) What is the minimum amplitude input that will ensure the transistor's remain
ing saturated over the full half of the sweep cycle?
(6) Sketch the current and voltage waveshapes.
(c) Calculate the sweep nonlinearity.
84. As an alternative to the sweep of Fig. 84, the squarewave excitation is
applied to the base as shown in Fig. 816. For the purposes of analysis assume that
each equivalent diode in the transistor has a forward resistance of 5 ohms and a
reverse resistance of 100 K. In addition, the collectortoemitter saturation resist
ance is 10 ohms. The coil parameters are L = 100 mh, R = 20 ohms, and C =
100 m&. (For this problem, remove the diode shunting the coil.)
(a) Sketch the current if the 100MSec square wave drives the transistor into
saturation.
240
TIMING
[Chap. 8
(6) How might this circuit be modified to linearize the sweep? Explain your
answer.
9+30 v
Fig. 816
86. In the switched sweep of Fig. 88o the plate resistance of the pentode, when
operating along the eb. m i n line, is 400 ohms. The coil used has an additional resistance
of 100 ohms and an inductance of 500 mh. Once the current builds up to 100 ma,
the tube is switched off (the plate voltage will not leave the e6,min line until this point).
E bb = 300 volts.
(o) Sketch the sweep current waveshape, specifying all times and time constants.
(6) Evaluate the sweep linearity.
(c) Repeat part a when the tube is not switched off until the current builds up to
its maximum value. Assume that the plate resistance in the active region is 100 K.
Calculate the peak plate voltage and give the time at which it occurs. Plot the
operating locus on the plate voltampere characteristics.
86. (a) If the coil of Prob. 85 has associated shunt capacity of 500 /i/xf, calculate
the value of Rd necessary for critical damping.
(6) Calculate the Rd that will result in a damping equation of the form
i = ke~ at cos fit
where a = fi.
(c) Plot the voltage and current recovery waveshapes for parts o and 6 on the same
graph. Which condition gives the best recovery response?
87. Assume, for this problem, that the coil capacity in the switched sweep of
Fig. 817 is zero. The plate resistance in the saturation region is r'„ = r p /(n + 1),
and this region corresponds to £i., m in < Ec.max.
(a) Sketch the voltage and current waveshapes to scale.
(6) Plot the locus of operation on the piecewiselinear tube characteristics,
(c) Prove that the tube remains conducting when the pulse first disappears.
y+300 v
100 mh
50 v
100 • x ™ 
A sec 420v
Fro. 817
LINEAR CUEEENT SWEEPS 241
88. A poweramplifier pentode having a saturation resistance of 200 ohms is used
to excite a 50mh 50ohm coil in the periodic sweep of Fig. 886. The maximum
saturation current of this tube, with zero controlgrid voltage, is 250 ma. E» =
400 volts.
(a) Calculate the optimum value of iJi if the coil current must vary from — 100 ma
to +100 ma over the sweep interval.
(6)' Sketch the plate current, diode current, and coil current to scale. The oscilla
tory recovery time is limited by external capacity to 10 per cent of the linearsweep
interval.
(c) Sketch the plate voltage waveshape.
(d) Plot the plate characteristic trajectory of this sweep.
(e) Calculate the sweep nonlinearity.
89. The coil of Prob. 81 is driven by the output of an emitter follower as shown
in Fig. 811. Assume that the output impedance of the approximately unity gain
stage is only 10 ohms. The peak current of the single sweep cycle is 200 ma, and this
circuit has a recovery diode and resistance critically damping the response. In addi
tion, shunting capacity is added to reduce the peak voltage across the coil to 100 volts.
(a) If a 25volt supply is used, what is the fastest possible sweep?
(6) Calculate the necessary drive for a single sweep of 5 msec duration.
(c) Sketch the voltage and current waveshapes to scale, paying particular attention
to the recovery portion of the cycle.
(d) Repeat part c when the diode is removed.
810. The circuit of Fig. 816 is used as a periodic sweep. The transistor has
a = 0.98, r, = 10 ohms, r b = 100 ohms, r d = 100 K, and the 10mh coil being driven
has a resistance of 10 ohms and 100 jujuf stray capacity.
(a) What is the highest frequency sweep possible if the recovery is limited to 10 per
cent of the sweep period? The peaktopeak sweep current must be 5 amp and
E kb = 30 volts.
(h) Calculate the drive waveshape, at the base of the transistor, required to pro
duce a periodic sweep of onehalf the frequency determined in part a. Ensure that
the transistor will be backbiased during the recovery interval. Calculate the
optimum value of R a . What function does this resistor perform?
811. A poweramplifier triode is used as a cathodefollower driver with the deflec
tion coil inserted in the cathode circuit. For this tube /» = 10, r p = 1.5 K, and the
plate supply is 400 volts. The same basic circuit is used for both horizontal and
vertical deflection in a television receiver. Both deflection coils have L = 50 mh,
R = 150 ohms, and C = 250 M/ if. For a single horizontal scan the current must
increase to 200 ma in 55 //sec with 8 jusec allowed for recovery. The vertical scan
requires a peak of 150 ma in 16 msec with 700 /isec left for the recovery.
(a) Specify the necessary trapezoidal drive signal for the horizontal sweep.
(6) Complete the design by calculating the diode resistance and recovery capacity.
(c) Sketch the circuit waveshapes.
(d) Repeat parts o, 6, and c for the vertical sweep.
812. The coil of Prob. 811 is excited from a generator having 200 ohms source
impedance.
(a) Specify the drives necessary to produce a 200ma peak 10msecduration
periodic sweep of the following geometry: a parabola, i = KP; an isosceles
triangle; a truncated isosceles triangle where the flat top is 3.3 msec long.
(b) Sketch the waveshapes produced on the oscilloscope screen if the coil is rotated
at 300 revolutions/sec and the sweeps are those given in part a.
813. Prove that the current bootstrap circuit employing only a single transistor,
242
TIMING
[Chap. 8
such as the one shown in Fig. 818, produces a sweep of poorer quality than the sim
ple switched coil of Fig. 84. (a = 0.98, r, = 10 ohms, n = 100 ohms, and r c =
1 megohm.)
L
i
A/W
it. R L , C
r— TSW — I
Rz
AAV
,1
T
Fia. 818
814. In the base amplifier of Fig. 814 the two transistors are identical, each having
= 50, r. = 10, n, = 200, and n = 100 K. The power supply Eu,  30 volts, and
R 1 = Ri = R t = 1 K. Plot the current gain as a function of R 3 .
816. We wish to use the circuit of Fig. 819a to produce a perfectly linear current
sweep i = kt after the switch is closed at t = 0.
(o) For this condition, sketch the voltampere characteristic of N, labeling all
slopes.
(6) If i = kt with N properly adjusted, what is the value of kl
(c) When N is represented by a twostage current amplifier with feedback (Fig.
8196), specify the necessary current gain /S in terms of Ri, r„ and tl
(d) How does the time constant of the circuit vary as a function of /3, for small
variations in 0, in the vicinity of the optimum value?
N
t "*!
: R 0\fih r.l
r B »R 1
(a)
(b)
Fig. 819
816. In the current sweep of Fig. 815 the driving voltage increases linearly from
to 10 volts in 100 /isec. The circuit components are Ri = 100 ohms, L = 500 mh,
Rl  50 ohms, C = 0.01 i^f, Eu, = 20 volts, a = 0.98, r. = 20 ohms, and r c = 500 K.
(a) Sketch the coil current and voltage to scale, specifying all values. Make all
reasonable approximations.
(6) Calculate the value of the damping resistance which should be included across
the coil if the maximum collector voltage must be limited to 40 volts. Repeat part
a, first when this resistor is directly across the coil, and then when it is in series with a
damping diode.
817. A series RL circuit is excited by a linearsweep voltage from a lowimpedance
source such as a cathode follower. The total series resistance is 500 ohms, and the
coil inductance is 1 henry. Evaluate and sketch the current and voltage waveshapes
appearing across L
(a) When t is long compared with the time duration of a single applied drive signal.
(6) When t is short compared with the time duration of the drive signal.
LINEAR CURRENT SWEEPS
243
(c) When the applied voltage has a peak amplitude of 100 volts and a duration of
100 /usee.
818. The triode current sweep of Fig. 820 is driven by a perfectly linear sweep
voltage of 100 volts peak amplitude and 50 /tsec duration. The tube used is a 12AU7.
(a) Sketch both the plate voltage and plate current waveshapes when the damping
diode is omitted. (Make all reasonable approximations.)
(b) Calculate the value of Rd necessary to limit the maximum platetocathode
voltage to 600 volts.
(c) Repeat part a when the diode is included and when Rd is given by the answer
of part 6.
+300
100 v
819. (a) Prove that the coil current in the sweep of Fig. 815 may be expressed as
i = Ae'* + Bl +C
where A, B, C, and t are constants of the circuit.
(6) Evaluate the above constants in terms of the circuit and transistor parameters.
(c) Is it possible to generate a perfectly linear sweep? What conditions must be
satisfied if the sweep nonlinearity is to be minimized?
BIBLIOGRAPHY
Chance, B., et si.: "Waveforms," Massachusetts Institute of Technology Radiation
Laboratory Series, vol. 19, McGrawHill Book Company, Inc., New York, 1949.
Goodrich, H. C: A Transistorized Horizontal Deflection System, RCA Rev., vol. 18,
pp. 293321, September, 1957.
Millman, J., and H. Taub: "Pulse and Digital Circuits," McGrawHill Book Com
pany, Inc., New York, 1956.
Soller, T., et al.: "Cathoderay Tube Displays," Massachusetts Institute of Teclu
nology Radiation Laboratory Series, vol. 22, McGrawHill Book Company, Inc.,
New York, 1948.
Sziklai, G. C, R. D. Lohman, and G. B. Herzog: A Study of Transistor Circuits for
Television, Proc. IRE, vol. 41, no. 6, pp. 708717, 1953.
PART 3
SWITCHING
CHAPTER 9
PLATEGRID AND COLLECTORBASECOUPLED
MULTIVIBRATORS
91. Basic Multivibrator Considerations. Any closedloop regener
ative system having two or more clearly defined stable or quasistable
states, each of which is maintained without recourse to external forcing,
might well be categorized as a multivibrator. The application of a low
energycontent stimulus, a trigger, starts the switching action which
drives the operating point away from one stable state and toward the
next. But the important attribute of the multivibrator which differ
entiates it from other multistate systems, such as a switch or relay, is
that once the trigger brings the cir
cuit into its regenerative region, the e . o_. i A ^ e _ Ae .
additional energy necessary to com
plete the transition is supplied by Sw
the system itself.
The necessary and sufficient condi
e
tions for any system to operate as a ^A^e m ^
multivibrator may best be estab „ „ , „ _. '
fished by considering the functional ^ ^ BaS1 ° re « eneratlve ^^
form of the multivibrator, shown in the block diagram of Fig. 91. At
least two active elements are assumed included in A, contributing the
necessary positive gain.
Upon the application of an input stimulus, e in , the signal returned
through the amplifier and feedback network /3 is
e'„ = /34e ln (9_D
When pA > 1, the returned signal is larger than the applied stimulus,
and once the switch is closed, it will reinforce the original signal. The
closedloop system is unstable, with buildup continuing until the balance
of returned signal and input is achieved, either through a decrease in the
amplifier gain (limiting) or by the destruction of the transmission path
(active element cut off).
Balance conditions require that when the switch is closed
< = 4. (92)
247
248
SWITCHING
[Chap. 9
For possible switching, the system must have more than one stable point
satisfying Eq. (92).
Figure 92, where we have superimposed the unity gain locus (e' = e in )
on the loop transfer characteristic of a typical amplifier, i.e., PAe^ versus
e iB , serves to illustrate the switching action. If the particular point
selected to break the system's loop, for the evaluation of this curve, is
normally restricted to positive or negative voltages (e.g., at the plate of a
tube), then the whole curve will be shifted along the balance line into
either the first or third quadrant. Inherent circuit limitations will
always account for the two horizontal regions.
Satisfaction of the unityloopgain criterion is a necessary but insuffi
cient condition for stable operation. Consider point Y (where e' a = e m
e' o °A0e in <■
Fig. 92. Loop transfer characteristics for circuit of Fig. 91.
= 0), which, at first glance, appears stable. The slope of the transfer
characteristic, evaluated at a particular point, is simply the value of the
corresponding loop gain &A. At the origin, $A > 1, and thus any slight
perturbation which might momentarily shift the operating point into
the first quadrant will be amplified and returned as a much larger dis
turbance. The operating point travels Up the transfer characteristic
from Y to point X, where the balance condition is again satisfied. Here
the incremental gain f3A is less than 1; it is in fact zero. Any small
disturbance away from X rapidly damps out, and the operating point
returns to X, one of the two stable points. Similar reasoning shows that
the other one is point Z.
We conclude, from the previous discussion, that the second necessary
condition for stable switching is that fiA > 1 in the active region and
that fiA < 1 at the stable point. It also follows that there always must
be a point of unstable equilibrium between each set of stable points. As
an exercise, the reader may furnish the proof of this statement.
Sec. 92] plategrid and collectorbase multivibrators 249
If the active elements were transistors instead of vacuum tubes, we
would modify the previous discussion and define the regenerative action
in terms of the essential loop current gain.
The initial start from any unstable point, such as Y, would be
caused by circuit noise or some other small perturbation. However,
in order to switch from one absolutely stable point to the other, we
must disturb the system at least enough to shift the operating point
into the regenerative region where &A > 1. This region is easily deline
ated by marking its boundary points (j}A = 1). We simply find the two
points on the characteristic with unity slope (K, L). Thus the trigger
size required for switching from X to Z becomes
e T > e K — e x
To shift back from Z, we must introduce a positive disturbance greater
than ZL.
When energystorage elements are included in the internal transmission
path, the exponential decay will eventually bring the circuit from the
limiting to the regenerative region. As a consequence, the multivibrator
switches between the points satisfying the balance conditions without
resorting to external triggers. But since the system remains at each
particular state for a welldefined interval, this only changes the stable
to a quasistable point.
The flat portions of the transfer characteristic arise as a direct result
of driving one of the active elements into saturation or cutoff. It follows
that if we modify the load of the active element by the addition of diode
waveshaping circuits (Chap. 2), multiple limiting regions become fea
sible. Each diode introduces an additional break into the transfer char
acteristics and may thus create a new stable, or if energystorage elements
are also included, a new quasistable, point.
Transition from state to state takes some small finite time interval,
primarily the time required to store or dissipate energy in the parasitic
elements. In addition, when the base amplifier contains transistors,
their onoff and carrier storage times will further affect the switching
speed. These delays are not germane to the system's basic behavior,
and we shall therefore ignore them in the following discussion.
92. Vacuumtube Bistable Multivibrator. The circuit of a vacuum
tube bistable multivibrator (bistable indicating two absolutely stable
states) appears in Fig. 93. Sometimes this circuit is also referred to as
an EcclesJordan circuit, a flipflop, or a binary. On being triggered,
each tube switches between two of its three possible states, full on, active,
or cut off; the particular two depend upon the adjustment of the circuit
parameters.
250
SWITCHING
[Chap. 9
It is easy to show that both tubes are normally in different states
except during the transition interval when they simultaneously become
active. Assume that both tubes are on, with E ce , a negative voltage,
chosen to place them in their active regions. Any slight disturbance,
such as noise, may momentarily raise the plate current of TV The
resultant drop in plate voltage is coupled by Ri, R 2 , and the speedup
capacitor C\ to the grid of T2. The crosscoupling reapplies the amplified
signal (now a voltage rise) back to the grid of T\, where it reinforces
the original perturbation. Thus the regenerative action drives T\ toward
saturation and T 2 toward cutoff. Switching ceases when one or both
tubes reach their limit.
Three possible models exist for each tube, one when the tube is cut off
(Fig. 94o), the second when it is in its active region (Fig. 946), and the
Fig. 93. Vacuumtube bistable multivibrator.
third when it is saturated (Fig. 94c). (Figure 94a and c are degen
erate forms of the general platecircuit model shown in Fig. 946.) In
drawing the model holding during saturation, we neglected the effect
of the slight positive grid excursion. When this term becomes significant,
the model of Fig. 946, with the appropriate positive value of e c , will be
used instead.
In attempting to analyze a multivibrator, a circuit where many
combinations of tube states are possible, where and how do we begin?
The answer is "anywhere"; any logical initial assumption serves as a
convenient starting point and, if incorrect, will eventually lead to a con
tradiction. One of the limited number of other choices will then be
the correct tube state. Since both tubes cannot remain in the same
condition, the only permissible combinations of tube states are those
tabulated below. We assume that the circuit has been triggered so
that Ti is the on (active) tube and Tt is the off tube.
Permissible States of Tubes
Full on Active
Full on Cut off
Active Cut off
Sec. 92] plategrid and collectorbabe multivibrators
251
Our object thus is to start, say, by assuming Ti fully on, and then
to carry the steadystatecircuit analysis through to its logical con
clusion. Usually (Ri + Ri) » Rl and the loading of the plate by the
coupling network may be ignored. From Fig. 94c,
And by superposition
Ed
Eui =
R, + r v Eli
Ri
■E„+ n R *
Eu
(93)
(94)
Rl ~\ 7t2
where E b ,i is given by Eq. (93). Cutoff for this circuit is
w — ^*»
Eia, —
Now if the actual grid voltage E c i is more negative than E a
Ti is cut off and Fig. 94a is the applicable model for calculation of
(95)
then
(e)
Fig. 94. Vacuumtube circuit models, (a) Cutoff region; (6) active region; (c)
saturation.
E bi and E cl . But if > E ci > E m , T2 is in its active region and we
must use the model of Fig. 94b. At the stable point both tubes cannot
be active; if T 2 is part on, then Ti must be full on, and our original
assumption as to the circuit state was correct.
To continue, suppose that in this circuit E e2 < E m (T2 cut off); then
it follows from Fig. 94o that
^2 ra 1 "i
E C ]
■ Ebt +
E e ,
(96)
.Ri f Rt Ri + R2
But to verify the initial assumption, namely, Ti full on, the solution of
Eq. (96) must be E ci > 0.
Gridcircuit limiting prevents the grid from becoming more than
slightly positive regardless of the solution of Eq. (96). In the model
used, the grid loading was omitted in order to simplify the calculations.
252
SWITCHING
[Chap. 9
The alternative solution of Eq. (96), E co < E \ < 0, contradicts our
original assumption, and therefore the problem must be repeated from
the now known tube states, T\ part on and T 2 cut off. Because Eq. (96)
was written for T 2 off, the substitution of the value of E e \ given by it
into the model of Fig. 946 enables us to resolve the contradiction and
find the correct values of e b i and e c2 .
The switching process, the injection of a trigger turning the off tube
on or the on tube off, will just interchange the states of the two tubes
of the symmetrical circuit of Fig. 93. In an asymmetrical multi
vibrator, all voltages must be recalculated as a completely independent
problem. The widest separation between the two stable states, and
consequently the easiest to distinguish, occurs when each tube switches
between full on and cut off. Under these circumstances the total plate
voltage swing becomes
Ae 6l
S a
Em, — Et,,
RlEh,
Rl + T p
(97)
Proper multivibrator operation is predicated on our establishing the
correct grid voltages in each of the two circuit states. Since this depends
Si
=E,
<?c2i0
Fig. 95. Circuit models under limiting conditions, (a) Both tubes saturated; (6)
both tubes cut off.
on E ce , Ri, and Ri, we should determine their interrelationships, at least
at the limits, as a guide in circuit design. One limit occurs when both
tubes are always saturated, the other when both are cut off. The circuit
models holding under the limiting conditions, and where (Ri + Ri) > Rl,
are sketched in Fig. 95.
In order to prevent complete circuit saturation, from Fig. 95o we
see that
Ri
E ce < 
Ri
Et,
(98)
where Eb,i is given by Eq. (93). The other circuit limit, found from
Fig. 95&, is expressed in Eq. (99).
Ecc> 
Ri
E»
Eu
(99)
Sec. 92] plategkid and collectorbase multivibrators 253
The above approximation holds only for high/* tubes. Generally, the
center value of Eqs. (98) and (99) represents a good choice in that it
allows for the greatest possible variation in the tube and circuit param
eters before the multivibrator will cease functioning.
Rather than switch the multivibrator from one state to the other by
first injecting a positive trigger to turn the tube on and then a negative
one to turn it off, we might much more conveniently apply a train of
+200
200 v
100
• Triggers  
(b)
Fig. 96. (a) Bistable multivibrator triggered by a negative pulse train; (6) resultant
plate waveshapes.
only positive or negative pulses. In order to maintain gridtogrid
isolation, the pulses are coupled into the tubes through diodes (Fig. 96o),
producing the change in state at the plate of T 2 shown in Fig. 966.
The negative pulse drives the on tube into its active region. The
positive amplified pulse appearing at the plate is coupled into the off
tube and pulls it into conduction. For positive trigger inputs, the diodes
are reversed.
Example 91. In this problem we shall investigate the role performed by the
speedup capacitor d in the bistable multivibrator of Fig. 96. The circuit is excited
by negative triggers injected into both grids through diodes. The tube used has
ft = 20, r, = 10 K, r c = 200 ohms, C„ h = 10 nrf, and C m = 10 n4
254 switching [Chap. 9
From Eqs. (93) and (97) we see that the plate voltage will swing from 200 volts,
when the tube is cut off, to 100 volts, when the grid is driven slightly positive. To
prevent false triggering, the grid of the off tube should be maintained somewhat below
cutoff, say 10 volts below the cutoff value of — 10 volts. Assuming that R x + R t »
Rl, then with the grid of the off tube at —20 volts, the drop across R 2 must be 80 volts
(Fig. 94c). Since the plate of the on tube is at 100 volts, the drop across R t will be
120 volts. Hence
Ri _ 120
Rt 80
and we can, quite arbitrarily, choose ijj = 0.5 megohm and Ri = 0.75 megohm.
We must now check that the on tube is saturated when these coupling resistors are
used. Substituting the circuit values into the model of Fig. 94o, the grid voltage of
Ti becomes
e n = 20 volts
Of course, this calculation neglected the positivegrid limiting and ea would actually
be only a fraction of a volt.
When the external pulse is injected, the cutoff grid of T 2 will charge from —20 volts
toward the new value established by the input trigger, with the time constant
t« = Ri  RiC ek = 2.85 fxsec
Once the tube becomes active, the input capacity seen looking into the grid becomes
C in = C, h + C„(l  A) = 10 + 10[1  (10)] = 120 ««f
Thereafter the charging continues with the much longer time constant of
t 6 = .Ri  Rid* = 34 Msec
If the input pulse is extremely narrow, the net change in the grid voltage may be
insufficient to maintain the switched tube states after the trigger is removed; the
multivibrator will subsequently return to its original condition.
By shunting Ri with C lt the coupling network is changed into the compensated
attenuator discussed in Sec. 19. Usually, for the fastest possible switching, it is
adjusted slightly overcompensated. However, for the purpose of this discussion,
assume that CiRi = RiC ia or that
C : = 80 vrf
This capacity is far from critical, and in an actual circuit it may vary from 25 to
200 niii. The switching time constants now become, from Eq. (134),
1 '* R, = 0.045 jusec
0\ + Crt
R, = 0.245 j«sec
where R, is the 5K source impedance seen looking into the plate of the active tube.
We see that the presence of Ci reduces the critical time constant by a factor of more
than 100.
The above argument attempted to show, in a relatively qualitative
manner, that Ci speeds up the switching process and permits operation
with quite narrow trigger pulses. Many factors are changing during
Sec. 93] plategrid and collectorbase multivibrators
255
the interval considered: one tube is turning off; the other tube first
becomes active and then its grid rapidly saturates. Therefore no attempt
will be made to define the required pulse width. We can, however, say
that it should be somewhat wider than the calculated time constant t£
but very much narrower than the pulse needed in the uncompensated
circuit.
The bistable multivibrator is a device widely used in digital computers
and in control equipment for counting. Each unit registers two counts :
the first input trigger turns off the tube, and the second one turns it
back on. Thus, if we differentiate the plate waveshape (Fig. 966), we
shall derive only one negative trigger for every two applied. By coupling
the output pulse to another bistable we count down by four, and finally
the cascading of n circuits allows counting by 2", yielding one output pulse
per 2 n inputs. Various feedback arrangements, which preset the cas
caded binary chain, will present a single output pulse for any desired
count.
93. Transistor Bistable Multivibrator. A collectorbasecoupled mul
tivibrator is the transistor equivalent of the plategridcoupled multi
vibrator previously discussed. The mode of operation is now determined
Fig. 97. Transistor bistable multivibrator.
by the base current flow, and therefore, for ensured regenerative switch
ing, the loop current gain must be greater than unity. If we keep these
specific features in mind, then this circuit is also amenable to the argu
ments employed in Sec. 92.
Assume that Ti, of the transistor bistable of Fig. 97, is in its active
region. (The circuit shown uses np^n transistors; for pnp transistors
all polarities are reversed.) Any increase in its base current causes a
proportional increase in collector current. But this is reflected as a
decrease in the base current of T 2 , where it is amplified and finally
reapplied as an increase in the base current of 2*1. The original current
change has thus been multiplied by the total loop current gain. Eventu
ally the circuit limits with either T\ full on or T 2 cut off. Since the
256
SWITCHING
[Chap. 9
speedup capacitor couples the full change of collector current into the
other transistor's base, the buildup process is extremely rapid. And
as in our analysis of the vacuumtube circuit, transition time will be
ignored.
A separate model might be drawn for each of the three regions of
transistor operation. If, as is normally the case, R c is very much smaller
than the transistor output impedance and if Ri  Rz is very much larger
than the small input impedance, then the models become extremely sim
ple. Figure 98a shows the model holding for cutoff, 986 the one defining
the active region, and 98c that holding under saturation conditions.
For the solution of this circuit, we shall start by assuming Ti cut off.
The model of Fig. 98o gives the base current of Ti as
IbZ
Eu
+
E c <
(910)
Ri + R c Ri
Models for the active and saturation region must agree at the bound
ary. From Fig. 98c, we see that the collectoremitter voltage of a
saturated transistor is zero. Substitution of this limit into the model
of the active region (Fig. 986) results in
or
= Eu, 
Ebb
(iibRc
i h >
PR.
(911)
Equation (911) expresses the bounding condition for maintaining a
transistor in saturation.
l&bb
>Rc
c
0h\Q
> Rl
bo—
e
[ Rz
Err
I
6o
(a) (b)
Fig. 98. General transistor models, (a) Cutoff region; (b) active region; (c) satura
tion region.
A comparison of Eqs. (910) and (911) determines the actual state of
T 2 and enables us to choose the appropriate model. E c i and hi or E iel
are now calculated, and the initial assumption checked.
If T\ is not cut off, the contradiction must be resolved by a new solution
from the other possible starting point, T\ active and r 2 full on. It is, of
Sec. 93] plategrid and collectorbase multivibrators
257
course, necessary to modify the general models of Fig. 98 so that they
satisfy the now known transistor states. In particular, we must connect
the base junction of the coupling network in Fig. 986 to ground when
solving for E cl and hi.
The limits of E cc or the coupling resistors are found at the two bound
aries, complete saturation and complete cutoff. From Eq. (910), the
minimum value of E cc for ibi > is
E a > 
Ri
Ri + R,
En,
(912)
The maximum value of E ce , above which both transistors are always
saturated, will be found directly from Eq. (911) through recognizing
that, under this condition, the total base current must be supplied by
E ec . Therefore
Rz
E c ,
< m Ebh
(913)
To ensure that one transistor is driven into cutoff while the other
remains in the active region, much more stringent limitations must be
placed on E cc [Eq. (914)]. If we
assume that T 2 is on and solve for
the condition that the voltage at
the base of T\ must be less than
we obtain
zero,
E„ <
R2 w
R[ Ec '
(914)
Positive trigger inputs
Fig. 99. Simple transistor multivibrator
using one power supply and switching
between full on and cutoff.
where E ci is the collector voltage
of the conducting transistor.
It should be noted that with T 2
full on {E c2 = 0), Ti is cut off for all
Ecc < [Eq. (914)]. By setting
E cc = 0, only one power supply is
required instead of two. Furthermore, since the multivibrator switches
between full on and full off, current never flows in R 2 and it also may be
omitted. The very simple multivibrator of Fig. 99 results.
The stablestate current and voltage values for this circuit, with T\
cut off and T 2 full on, are
in — %i =
e„2 = e c i =
En
R c + R
REtb
R c + R
258 switching [Chap. 9
But in order to ensure the saturation of T*, t&2 must satisfy the condition
given in Eq. (911):
. _ Etb Ebb , a . .
l "  wtr  m (9_15)
The solution of Eq. (915) establishes the maximum value of B as
R < (P  l)Rc (916)
By treating this circuit as a current amplifier, we shall now demonstrate
that the condition described above is the very one necessary for guaran
teed regenerative switching. Suppose that the loop is broken at the base
of Tx and that point f is shorted to ground. Then the loop current gain,
from an input of T\ to point f , is
£ = A < = (m) v (9  17)
For regenerative operation A c > 1, and thus
R < (/3  l)R e
which is identically the condition previously given by Eq. (916). A
similar inequality must also be satisfied in. the first multivibrator dis
cussed in this section.
The change of circuit state, on to off and back on again, is effected by
positive triggers injected into both bases through diodes. They drive
the off transistor into the conduction region, and the resultant of the
amplified pulse and the original triggers turn the on transistor off. Nega
tive triggers will also work, but the diodes would have to be reversed.
Collector waveshapes, except for voltage values, are of identical form
with those of the vacuumtube multivibrator of Fig. 96. The maximum
swing will, of course, be much smaller, and the minimum voltage value
can now go to zero, but the two circuit states are still clearly delineated,
one at zero and the other close to I?».
94. A Monostable Transistor Multivibrator. The introduction of a
single energystorage element in the regenerative transmission path
creates a circuit having one stable and one quasistable state. If we
examine the resultant monostable circuit of Fig. 910, we note that
since the base of T 2 is returned to En, this transistor must be on, either
in its active region or saturated. The polarity of E cc is always opposite
to that of Eu, and it is relatively easy to ensure that 7\ will be cut off.
In the particular multivibrator shown, pnp transistors are used as the
active elements. Em, will therefore be negative, and E cc positive. If
Sec. 94] plategrid and collectorbase multivibrators
259
npn transistors were used instead, all voltages and currents would be
reversed.
The quasistable circuit state, Tz cut off, is contingent on the charge
in the coupling capacitor maintaining e^ > 0. However, a discharge
path exists and, regardless of the initial condition, C must decay with
eui subsequently reaching zero. At this point Tt turns back on. Regen
eration turns Ti off, and the multivibrator is back in its one "mono"
stable state.
Analysis of this circuit will proceed on the assumption that it is
designed for the maximum possible collector voltage variation, i.e.,
both transistors driven between saturation and cutoff. If any particular
answer does not support this contention, then the problem must be
resolved by drawing new models that will yield consistent results.
Fig. 910. Collectorbasecoupled monostable multivibrator using pnp transistors.
Eu is negative and E, c positive for proper biasing.
The stable circuit conditions are (from the models of Fig. 98)
T 2 on:
c c i = Ebb
e C 2 =
e b ei(0r) =
42(0) =
Ri \ Rz
Ebb
R
E cl
If Ti is to be saturated, t» 2 must also satisfy Eq. (911), leading to the
following restriction on R :
R < 0R C (918)
If this inequality is not satisfied, then, in its normal state, T 2 will be
active instead of being saturated.
A negative trigger injected at the base of 7\ turns this transistor on,
and the change in its collector voltage is immediately coupled through
C to the base of T 2 , turning it off. The circuit has now entered its quasi
stable region, and the new models needed to define operation are those
given in Fig. 911.
260 switching [Chap. 9
We determine the time elapsed before the circuit can return to its
stable state from Fig. 91 la. As the righthand side of C charges from
31
6i
t c i
c
b 2
o—VWi
R
C 2.
e 6«2
■Efcfc^
i
R c
«CZ
R 2
(a) (b)
Fig. 911. Circuit models holding under quasistable conditions, (a) Collector of Ti
to the base of Tt; (ft) collector of Ti to the base of 7*1.
its original value of — En, toward En,, the equation of the base voltage of
Ti becomes
e M (t) = E»  2JS?»e" T » n = RC
At i = h, e b 2 = 0, and T 2 again conducts. Substitution of the boundary
value into the equation defining en(<) results in
h = ti In 2
(919)
To guarantee the saturation of Ti during the unstable period, the follow
ing relationship must be satisfied [from Fig. 9116 and Eq. (911)]:
_ En, , Eec > En,
R c + Ri Ri PRc
(920)
After the circuit reswitches and recovery begins, the models of Fig. 911
are no longer valid. We must consider the new problem posed in Fig.
912, where C recharges toward Eh, from its initial value of zero. The
collector of Ti recovers toward its
<?! *°~ b z stable state, also Ew, with the recovery
I *^f*  l» time constant T2 = R C C
1 " e cl Within four time constants, re
covery is virtually complete. Since
we wish the recovery time to be small
compared with the output pulse width,
R c <K R. This multivibrator is also
open to simplification by omitting R% and E cc just as discussed in Sec. 93.
Their presence aids in maintaining r 2 well below cutoff while the circuit
is in its normal state and consequently prevents false triggering by small
noise pulses.
Circuit waveshapes are shown in Fig. 913, and we might note that
the output pulse having the best shape appears at the collector of TV
I
i
Fig. 912. Recovery circuit.
Sec. 95] plategrid and collectokbase multivibrators
261
Since this point is isolated from the single RC timing circuit, external
loading, introduced by the coupling to the next stage, will not affect
the pulse duration. If we differentiate the output pulse, the resultant
t
B,+fl 2 Ecc
t
eel
c
tOE bb t
t
t
E bb
Fig. 913. Monostable multivibrator waveshapes for the pnp circuit of Fig. 910.
positive trigger is delayed from the applied input by the pulse duration t\.
Therefore the monostable multivibrator may be used as either a pulse or
a delay trigger generator.
95. A Vacuumtube Monostable Multivibrator. Even a perfunctory
inspection of the monostable of Fig. 914 indicates that in its normal state
+200
=r=C,
Fio. 914. Vacuumtube monostable multivibrator.
T 2 is fully on (the grid returned through R to Ey, will be slightly above
zero, and the plate will be slightly below its saturation value of 100 volts).
Proper choice of E m will bias Ti well below cutoff and yet allow it to
switch fully on when the circuit is triggered into the unstable state.
262 switching [Chap. 9
If we neglect both the small positive grid voltage and the loading of
Rl by the resistivecoupling network, the grid of the off tube will be at
Rt in i Ri
e„i(0)
E b ,+
E c ,
20 volts
(921)
R\ ■+• Ri Ri ■+■ Ri
which is below the cutoff value of — 10 volts. This tube's plate is, of
course, at Em,. The only other unknown initial condition is the voltage
across C, which remains invariant across the transition from the stable
to the unstable state. Since one end of C is returned to ground through
(a) (b)
Fig. 915. Quasistable state circuit models, (a) Plate of 7\ to the grid of T s ; (6)
plate of r 2 to the grid of T,.
the conducting grid of 7 1 2 and the other end is connected to the plate
of the off tube,
e,(0~) = e„i(0)  e c ,(0) Si E»
After a positive trigger at the grid of T\ switches the circuit into its
quasistable state, the models defining the behavior become those shown
in Fig. 915. Figure 915a is the more important of the two because it
contains the circuit's single energystorage element. The model can be
further simplified by replacing the tube with its Thevenin equivalent —
a 100volt source {E hB ) together with a 5,000ohm resistance (Rl \\ r p ).
This approximation neglects the slight positive grid excursion that will
lower the plate to somewhat below Et,. By using the reduced circuit
we can now solve for the grid voltage of T*.
2Ebb — Et,
e* 2 (0+) = En, 
R
(922)
R + Rl\\ r p
But since R~2> Rz,\\r p
e c2 (0+) Si (#«.  Et.) = Si = 100 (923)
where Si, the total plate voltage swing of Ti as it switches from cutoff to
saturation, is also given by
Rl
Si =
r p + Ri
Em
(924)
The grid starts recovering from — 100 toward 200 volts with the time
constant
n = (R + Rl II r„)C S RC = 56 Msec
Sec. 95] plategrid and collectorbase multivibrators
263
Once it reaches the cutoff value of — 10 volts, T t becomes active and the
circuit regeneration will rapidly turn T\ off and T% fully on. Conse
quently, the duration of the quasistable state, as evaluated from the
exponentialresponse equation, is
t\ — t\ In
Ebb + <Si
E» +
Eu
20 /isec
(925)
Since, at least to a first approximation, all the voltage terms of Eq. (925)
are linear functions of Ebb, we expect that the pulse duration will be
relatively independent of any variations in the plate supply voltage.
Both n and r P will vary with the current flow, n only slightly but r p
drastically. As a consequence Si will vary somewhat with E bb , and so
will the pulse duration.
The final charge on C at the instant before switching back to the stable
state is
e t (h) = E b . + — = 110 volts
This value must be used to calculate the initial value of the recovery
waveshape from the model of Fig. 916. Since the transition occurs at
■=E,
(a) ~  <b)
Fig. 916. Recovery models for plategridcoupled monostable multivibrator, (o)
pi to gs; (6) ps to pi.
— Ebb/ ii instead of at zero, during the switching process the tube goes
directly from cutoff to saturation. As a result, jumps appear at both the
grid of T 2 and the plate of Ti. From Fig. 916o,
E,
c2,n
but since r c <K R L ,
E,
= Ebb  (E h . + Ebblv)
Rl + U
Tk( Sl ~ir) = 2  2 volts (9_26)
We might note that the voltage term in Eq. (926) is simply the total
change of loop voltage due to the change of the circuit state and that it
divides proportionately across r e and R L  The total jump at the grid of
12.2 volts (from 10 to +2.2 volts) will also appear at the plate of T ly
264
SWITCHING
[Chap. 9
raising its voltage from 100 to 112.2 volts. After the jump, recovery is
rapid — back to the initial steadystate value with the fast time constant
of
r% = (Rl + r c )C^ R L C = 1 Msec
To obtain the recovery waveshapes at the plate of T 2 and the grid
of T\ we must treat the model of Fig. 9166. These are most easily
„ 200
t "
t
e cl
Bci(O)
10
>
20
200 v
100 v
^
78 v
22vj
S** 1
t
12.2 V
toE bb
r^i2 »
10 v
A£ c2 
l..^ *
100 v
■E>2 mm
"c2 max
o
£»
Si
Fig. 917. Waveshapes of plategridcoupled monostable multivibrator of Fig. 914.
found by imposing the effect of the positive grid voltage on the quiescent
values. The 2.2volt grid excursion is amplified (A = — 10) and reflected
into the plate of T 2 , causing the voltage to drop 22 volts below the nom
inal value of 100 volts:
hRl
Eb:
— Ebs —
■E,
78 volts
(927)
t v + Rl
The plate now recovers toward E h , at a rate controlled by the grid circuit
charging. In addition, this large drop is coupled through the speedup
capacitor to the grid of 2\ and can only aid in driving it to well below
cutoff.
Sec. 95] plategrid and collectorbase multivibrators 265
Equation (927) may yield a negative answer for J5 iimln . Obviously,
this must be rejected since the model representation has led us to an
impossible solution. Once the tube is driven into the highgridcurrent
region, both r, and r c change markedly from the constant values assumed
in drawing the piecewiselinear models, and these variations, if taken into
account, would resolve the contradiction.
The waveshapes shown in Fig. 917 are of the same general form as
those found for the transistor monostable, and if we had accounted for
the small base region (about 200 mv), the transistor circuit also would
have exhibited some very small overshoots. The excursion below E b „
Fig. 918. Firingline intersection by charging exponentials,
appearing at the plate of T 2 , can be easily eliminated by incorporating a
plate bottoming diode which will conduct and limit the plate voltage to
some value slightly above Eu
As an alternative to the circuit of Fig. 914, we might return R to
ground instead of to Ew The switching sequence still remains the same,
with the same size plate swing and time constant; the grid, however, now
charges toward zero. The conduction point —Eh/p is not affected by
this circuit change, but since the multivibrator switches state closer
to the final value, the pulse duration will be much longer than that given
byEq. (925):
a
T\
In
Etb
(928)
and for a high/* tube, which fires almost at zero, this sweep takes approxi
mately four time constants.
Suppose that we adjust the RC time constant of the particular multi
vibrator having its grid returned to zero so that its period is identical
with that of the original circuit (Fig. 918). The curve charging toward
zero (1) crosses the — 2J»/m line at an extremely shallow angle, in fact
almost horizontally. This is in sharp contrast to curve 2, which expo
nentially charges toward Ebb and which intersects the firing line closer
266
SWITCHING
[Chap. 9
to the perpendicular. Any noise introduced into the system effectively
lowers the cutoff line to the dotted line of Fig. 918. Curve 1 fires
quite a bit prematurely (at <„), while curve 2 is only slightly affected
(fires at fe). Thus we conclude that by returning the grid to the highest
possible voltage, the pulse duration is made much less susceptible to
random disturbances.
96. Vacuumtube Astable Multivibrator. The insertion of a second
capacitor in the transmission loop, i.e., capacitive coupling from each
plate to the other tube's grid, makes both circuit states quasistable.
Figure 919 shows the circuit of the resultant astable multivibrator.
If both tubes are assumed to be in their active regions, regeneration will
rapidly drive one into cutoff and the other to saturation. This state is
unstable, lasting only until the particular capacitor that is keeping the
tube cut off discharges, at which point the switching action reverses
c 2
X
Fig. 919. Plategridcoupled astable multivibrator.
the tube states: the off tube goes on and the on tube goes off. The
astable multivibrator is selfstarting and freerunning.
We might note that the timing network coupling each plate to the
next grid is exactly the same as the single timing circuit of the mono
stable multivibrator (Fig. 914). The astable multivibrator may well be
considered as two monostable circuits operating sequentially, one trig
gered off whenever the other one turns on. The previous techniques are
readily applicable, and by starting anywhere we shall, within no more
than 2 cycles, reach the final periodic solution.
Since the exponential timing after the change of circuit state depends
upon the conditions existing immediately before switching, probably the
best starting point is where T 2 is full on, just going off, and T x is full off,
just going on. At this instant the model holding (Fig. 920a) is identical
with the one defining the stable state of the monostable multivibrator
(Fig. 915a). In our initial analysis, so as not to confuse or conceal the
basic simplicity of the behavior of this circuit beneath the mathematical
manipulations, we shall ignore the effects of any positive grid voltage.
In fact, we may even limit e c < by simply setting r c  0. The initial
Sec. 96] plateghid and collectorbase multivibrators 267
circuit voltages, from Eqs. (921), (922), and the assumed starting point,
are
—En
e c i =
M
It
Ri
e.i = ■£ E», &
en = E»
ebs = Eui
Tpi
r P 2 + Rl
En
These values determine the initial charge across each coupling capacitor
and therefore enable us to calculate the voltages at each tube element
just after switching. The models holding appear in Fig. 920 with the
charge indicated.
i — wv
— E u
Pi
C,
£*.+
I
1
(b) • r
Fig. 920. Astable circuit models (Ti on, T t off), (a) Sweep model j)i to g,; (b)
positive grid recovery model p« to g\.
As the circuit changes state, the plate voltage of the formerly off tube
Ti drops from E» to E M . This swing Si is coupled by C 2 to the grid of
T* and will cut off that tube. Since R* » Rl, the effect of the capacitor
charging current in determining the plate swing is insignificant. Figure
920a shows the timing circuit holding during the unstable interval.
The grid of T 2 recovers from — Si toward En, with the tube turning
back on when it reaches —E^/y The time required is
ti — Ti In
Ey, + Si
En, + Etb/n
Ti = Rid
(929)
When the multivibrator reswitches, the sequence of events repeats in
the network coupling the plate of Ti to the grid of TV
ti = n In
Eu + Sj
Ebb + Ebblp.
ti Si RiCi
(930)
The two waveshapes generated are displaced in time from one another,
one starting when the other ends (Fig. 921). Thus the total period is
the sum of the times given in Eqs. (929) and (930). If a squarewave
output is desired, the circuit is made completely symmetrical and h = U.
Up to this point we have concerned ourselves only with the circuit
behavior during the off period and we have ignored anything that
happened at the grid of the on tube. When T t switches on (Fig. 920b),
268
SWITCHING
[Chap. 9
the change in the grid circuit and the introduction of a finite value of
r c produce a positive grid jump [Eq. (926)], with a subsequent recovery
back to zero.
e,i(<) ^ tf.we"*
RlCi
(931)
Normally, r 3 <3C t 2 and the recovery of Ti is completed within a very
short period compared with the off time of T 2 . The grid jump also
Fig. 921. Waveshapes appearing in the plategridcoupled astable multivibrator.
appears amplified at the plate of T t and helps drive the other tube
off. Thus two driving source components must be considered in calcu
lating the exact response of the grid circuit of T 2 , the simple plate drop
from Eu, to E b ,i (previously considered), and the amplified exponential grid
decay of Eq. (931).
Since the positive grid recovery of Tx is assumed complete while the
grid of the off tube is still well below the cutoff value, there will be but
slight modification needed as to the actual duration of the off period.
Sec. 97] plategrid and collectoebase multivibbatoes
269
It seems that a reasonable approach is to simply amplify the grid recovery
waveshape [Eq. (931)] and superimpose it on the sweep waveshapes
which were found by ignoring the positive grid jump. This is indicated
by the broken lines of Fig. 921. If we should require a more exact
answer as to the sweep duration, we would have to solve for the response
of the appropriate platetogrid RC circuit to the applied driving function,
— Si + Ae c i(t), where e c i(t) was defined in Eq. (931).
97; Transistor Astable Multivibrator. The circuit of a symmetrical
astable multivibrator using npn transistors appears in Fig. 922. Each
transistor sequentially switches between full on and full off and behaves
in a manner similar to the single unstable state of the monostable multi
vibrator. The circuit waveshapes are also basically those of the mono
stable (Fig. 913), with the alternate transistor states displaced by half
of the total period (Fig. 923).
(350
Fig. 922. Transistor astable multivibrator.
Once T2 goes full on, the base of T\ is driven to —10 volts and the
collector of Ti recovers to Em, with the time constant R C C. To ensure
complete recovery before the circuit reswitches, the half period, iden
tically that given in Eq. (919), must be longer than the recovery time.
RC In 2 > ±R C C
(932)
If, in addition, we should want to ensure that the on transistor remains
saturated over the complete half cycle, then a further limitation, defined
by Eq. (918), must also be imposed on R. Combining Eqs. (918) and
(932), we see that R should be restricted to the range
PR,> R> 5.8R C
(933)
In the circuit of Fig. 922, R must lie between 5.8 and 50 K. Cur
rently available transistors have /3 values of 50 or more; hence the
limitation on R is not particularly severe. Since C is the same in both
charging paths, the larger the size of R, consistent with Eq. (933),
the smaller the percentage of the half period devoted to the collector
recovery exponential. The waveshapes of one transistor are shown in
270
SWITCHING
[Chap. 9
Fig. 923; those at the other transistor are identical but displaced by a
half period.
1 u
'61
En,
i Ebb
cl
(
_J
£
1 1
1 1
1 1
1 1
=.fl c C
1 /r
1/
1/
*
Fig. 923. Waveshapes of transistor astable multivibrator (taken at transistor T t ).
Example 92. By setting R  75 K in the circuit of Fig. 922, Eq. (933) is not
satisfied and the transistors switch between their cutoff and active regions. This
mode of operation is somewhat different from that previously considered for the
transistor, but since the basic behavior remains the same, we may use the standard
method of analysis.
As our arbitrary starting point, we shall assume that Ti is off, on the verge of switch
ing on, and that T 2 is in the opposite state. In the initial discussion the contribution
of the charging current through Ct and the base of T t will be neglected, and we shall
further assume that the total bias current is Eu/R. When Ti turns on, its collector
falls from 10 volts to
•Eci(0+) "Em, piiRc =» 10  0;4t? 1K= 2.5 volts
to Iv
The base of T 2 falls by the same amount and immediately starts charging from
—7.5 volts toward 10 volts with the time constant
ti = RC = 75 msec
Since the turnon point is zero volts, all the information necessary to calculate the
switching time is now known. The half period is
h  75 In 10 + 7 5 S 42 msec
In the previous calculations we neglected the component of the base current of T>
contributed by the recharge of Cj. This current flows through the 1K collector load
resistor, starting from a peak of
, 7.5 volts „ ,
im = 5 = 7.5 ma
tic
and decays toward zero with the fast time constant
T 2 = RcC = 1 msec
The extra current flow drives the transistor into saturation for a portion of the
cycle. This phenomenon is almost identical with the positive grid jump in the vac
Sec. 97] plategrid and collectorbase multivibrators
271
uumtube circuit where the fast overshoot brought the plate voltage below Et,. Just
as in that case, Tt<H n and the extra recovery exponential will not influence the sweep
timing. It may simply be superimposed on the normal sweep as shown by the heavy
broken lines in Fig. 924.
10
2.5
t
f
A
i
•
42 msec
! !
i i
* i
t
:
a
s
.
A
1
r
7.5
10
Fig. 924. Waveshapes for the multivibrator of Example 92.
Under the conditions of this problem, the initial drop, and hence the half period, are
dependent on the value of 0. A 10 per cent reduction, from 50 to 45, reduces the size
of the voltage drop to 6.75 volts and the sweep time to 39 msec. If the transistor
switched between cutoff and saturation, as it does when generating the waveshapes of
Fig. 923, then the period is determined solely by the BC network and not by the
parameters of the active circuit element.
Incomplete Circuit Recovery. We now propose to consider the addi
tional complexity introduced in calculating the duration of the half
period, if the collector recovery is not complete when the multivibrator
switches states. The symmetrical astable is relatively easy to treat,
once it builds up to periodicity, since the waveshapes of both transistors
are identical. The base of one transistor is driven off by the drop in the
collector voltage of the other transistor as it switches full on. At the
end of the unknown half period h, the collector voltage has risen only
from zero to the value found from the recovery exponential [Eq. (934)].
(Also see the waveshape in Fig. 925.)
«.(*i) = Ml  e''"') t 2 = R C C
(934)
Therefore the off transistor is driven only to — e c (h) instead of to — E».
The equation denning the rise of base voltage becomes
e h {t) = E»  [E* + e.(«i)]« rM * 1 n = RC
(935)
A change of state again occurs when e&(<i) = 0. But the elapsed
time depends on the initial drop [Eq. (934)], which itself is determined
272
SWITCHING
[Chap. 9
by the unknown half period. The interrelationships of the two equations
are illustrated in Fig. 925, especially by comparing it with Figs. 923
and 924.
The above argument has furnished us with two transcendental equa
tions [(934) and (935)] which must be simultaneously solved for the
unknown half period h. One standard method of solution is to graph
e e i as a function of various values of ti, as individually found from each
equation. The intersection represents the unique solution for both e cl
and <i.
„
e c (h)
^*~
,.>
/
t
«cl
Az
AE cl
f
i !
't
o
« t x ,
i
■p <! >
t
t
e M
AE cl
/
/
e c (h)
~Ebb
^
f
Fig. 925. Collector and base waveshapes due to incomplete recovery.
In the special case of a symmetrical transistor multivibrator, by sub
stituting Eq. (934) into (935), we find that the half period h corresponds
to the solution of
1 _ 2e«' /T ' + e '' /r 'e'' /T » = (936)
By making the following additional substitutions,
XT2
x = e
,hhi
Eq. (936) reduces to
2x + 1 =
x < 1
(937)
The restriction on x must be imposed so that the solution of Eq. (937)
for the half period will correspond to real values of time.
In general, X is not an integer and Eq. (937) would have to be solved
by numerical or graphical methods for the single root lying within
< xi < 1
Sec. 97] plategrid and collectorbase mtjltivd3Rators 273
Finally, since xi = exp [ — (ii/n)], the half period is
ii = n In —
X\
For X large, the recovery time constant ti is very much smaller than
that of the rise and the root of Eq. (937) is located at
This simply yields the half period for the complete recovery waveshape,
i.e., the maximum possible sweep duration of
h = n In 2
which was given in Eq. (919). The second limiting location of the tim
ing root appears where r\ = t 2 (i.e., when X = 1). Under this condition,
Eq. (937) may be expressed as
(x  l) s =
and the half period is reduced to zero.
1.0
,
0.8
k 0.6
0.4
0.2
0.2
0.4 0.6
1 r 2
0.8
1.0
Fig. 926. Normalized half period as a function of the ratio of time constants [Eq.
(937)].
A plot of the solution of Eq. (937) appears in Fig. 926. We can see
that the recovery exponential has almost no effect on the sweep period
when r 2 < 0.2ti, but as t 2 approaches the sweep time constant n, the
sweep time rapidly decreases toward zero. Operation in this mode is
usually avoided since the slightest variation in circuit parameters has
pronounced repercussions on the waveshape produced. There are multi
vibrators, however, which depend on the incomplete recovery behavior to
generate extremely fast pulses. In this case, the transistor would
never be allowed to saturate (Sec. 99) and the analysis would be slightly
more complicated than that given above.
If the multivibrator is asymmetrical, we shall be faced with four
transcendental equations, two for each half cycle. The evaluation of the
274 switching [Chap. 9
two unknown time intervals and the two starting voltages would clearly
be most tedious.
Returning briefly to the vacuumtube astable, incomplete recovery
also means that the double exponential grid waveshape, shown by the
broken lines of Fig. 921, persists until the off tube turns on. The plate
voltage of the on tube reflects the positive gridrecovery exponential
into the circuit which maintains the next grid cut off. Even in a sym
metrical circuit, each of the two equations would involve doubleenergy
storage conditions and would be extremely difficult to solve.
98. Inductively Timed Multivibrators. In the capacitively timed
circuits discussed above, the regenerative loop was interrupted and the
quasistable state established by cutting off the tube (transistor). This
(a) (b) (c)
Fig. 927. (o) Inductively timed monostable multivibrator; (6) model holding during
the stable state ; (c) model defining the timing interval.
action was voltagecontrolled, with the amount of stored energy and the
discharge path determining the duration of the pulse generated. As an
alternative, the loop gain can be reduced to below unity by keeping the
transistor saturated for a controlled time interval. Because this is a
currentdependent action, we shall use an inductance as the timing
element.
Let us consider the monostable multivibrator of Fig. 927a, where,
in the normal state, T\ is saturated. Since its collector voltage is zero,
T 2 must be cut off. The initial circuit conditions, from the model of
Fig. 9276, are
Ebb
/n(O) =
E cl (0r) =
E c2 (0)
R 3 + R
/ci(O) =
Ebb
Ri
E bl (0) = E M (Qr) =
Ri
Rs + Ri
Ebi
(938a)
(9386)
(938c)
(938d)
Sec. 98] plategrid and collectorbase multivibrators 275
and to ensure the saturation of Ti, fiRi > (R* + Ri). Because Ri S> R%,
the drop across R t may be neglected in any computations.
The injection of a negative pulse at the base of T x turns off this tran
sistor. The current previously flowing through L must remain constant
across the switching interval. Il(0~) will now flow into the base of T it
driving it far into saturation. The current in L immediately starts
flowing through the path shown in Fig. 927c, decaying toward
Ehb (939)
ill ~\~ Ri
with the time constant n = L/(Ri + R^). For regenerative switching,
the circuit must enter the active region by itself. This requires that
/„, given in Eq. (939) , must be less than the saturation value of the base
current of T 2 ; i.e., it must be below
/., = H (940)
By writing the required inequality, the condition which must be satis
fied may be expressed as /3fl 3 < Ri + Ri
Once T% enters the active region, the increase in its collector voltage
forwardbiases jTi, bringing it from cutoff into conduction. Regenera
tion completes the switching, carrying Ti into saturation and turning
T 2 back off.
The duration of the pulse generated may be found from the exponential
charging equation. Substituting the initial current [Eq. (9386)], the
final current [Eq. (940)], and the steadystate current [Eq. (939)], this
time becomes
h = rx In Z " ~ /c V ( °" ) 041)
The various current limits are indicated in the waveshapes of Fig. 928.
Care must be taken to limit the maximum voltage appearing at the
collector of T\. On starting, when the coil current switches from the
collector of T"i to the base of T 2 , this voltage jumps from zero [Eq. (938c)]
to
e«i(0+) = J.i(0)fl, =  2 E*
In general, Ri > R t and the peak collector voltage will be several times
as large as E».
At the end of the pulse, the inductance must recharge to its initial
state (the saturation current of Ti). Since Ti is again saturated (Fig.
9276), it does so with the relatively long time constant t% = L/Ri,
276
SWITCHING
[Chap. 9
and because the recovery time is much longer than the pulse width, this
circuit is only applicable where the wide pulse spacing can be tolerated.
Thermal instability in a transistor multivibrator is due to the tempera
turedependent I c o, which also flows into C, increasing the total charging
current. This reduces the sweep duration, and when attempting to
generate longduration pulses, the large resistance used in the capacitive
timed circuit compounds this instability. On the other hand, if an
inductance is used as the timing element, the series resistance must be
■Ri
Ehh
\ E bb
e C 2
Em
l f
iss
X
^■^
t
\ t
\5
^«—  " —
c
't
Fio. 928. Inductively timed monostable multivibrator waveshapes.
minimized if we wish to have a long time constant. The effects of I c o
become almost negligible and pulse time stability is ensured.
Inductive Astdble Operation (?). It is quite difficult to design a simple,
easily controlled, astable version of the inductively timed multivibrator.
In the symmetrical circuit of Fig. 929, depending solely on the LR
charging (the coil capacity assumed completely negligible), the time
constant which maintains the saturation of one transistor is
Tl =
Ri + R%
But the coil recovers with the much longer time constant
L
Tj =
Ri
Sec. 99] plategrid and collectorbase multivibrators 277
The argument employed with respect to the incomplete recovery in the
capacitively timed circuit, which led to Eq. (937) , also holdsf or this circuit.
We draw the conclusion that no root corresponding to real time can be
found for the condition of n < t 2 , and consequently the period of this
multivibrator must be zero.
In the special case where the coupling resistor Rz is set equal to zero
and where Ri is very small, the transistor parameters may play a major
role in determining the two time con
stants of the circuit. During the satu
ration interval, the conduction path is
through the basetoemitter resistance
of the transistor. Recovery is through
the much larger saturation resistance
of the collector emitter path, and there
fore with a faster time constant. Thus
r2 < t\, and the period may be found
by solving an equation similar to Eq.
(937). It should again be noted that Fig. 929. Symmetrical circuit with
this multivibrator depends for its tim inductive energystorage elements
ing on the secondorder effects of the (not an astable circuit) '
circuit and its operation may not be very dependable.
Several freerunning circuits exist that apparently use inductive tim
ing for their operation. In all cases, either a second energystorage
element is present, making the coil a resonant circuit, or the two coils are
coupled and the core driven into saturation on alternative half cycles.
These circuits depend for their timing on other than a single mode of
energy storage. Their solution is somewhat more complicated than the
simple multivibrators discussed and is therefore beyond the scope of this
section of the text.
99. Multivibrator Transition Time. A number of approximations
were made in the course of our analysis of the multivibrator in the
interest of keeping the important characteristics of the circuit in the fore
front. The major assumption was that the multivibrator switched
states in zero time, which allowed us to treat only the circuit behavior on
both sides of the boundary. Calculation of the exact switching interval
is much too complex for a simple piecewiselinear approach, especially
since many of the significant terms were thrown out in linearizing the cir
cuit. A qualitative discussion will still point out the problems involved
in the switching process and will lead to the necessary conditions for
optimization.
The transition time of a vacuumtube multivibrator is primarily lim
ited by the parasitic capacity and inductance present in the circuit. For
ideal operation, the change of tube state, i.e., from on to off, should be
278 switching [Chap. 9
accompanied by large instantaneous changes in both tube current and
voltage, but lead inductance limits the time rate of change of current and
stray capacity will prevent any instantaneous change of circuit voltage.
The response to a change of state is quite similar to the transient response
of the tubes and coupling networks to an applied unit step of grid voltage.
For switching rates up to about 100 kc this is not very important; but
if we try to design multivibrators to operate at several megacycles, the
transition time is often the limiting factor.
The conclusions to be drawn from the above discussion are that the
switching time may be improved by using small plate load resistors, by
keeping the stray capacity low, and, if necessary, by highfrequency
compensation. Pentodes, with their improved highfrequency response,
are also occasionally used in highspeed multivibrators.
Transistor multivibrators normally operate with very small values of
collector resistance, which makes the effects of the parasitic elements
relatively unimportant. The basic limitations on switching time are
due to the properties of the transistor itself. Three factors must be
considered :
1. The transit time in the transistor
2. The cutoff frequency of the transistor
3. The storage time of the minority carriers
Current carrier velocity in a transistor, or any other semiconductor,
is quite slow compared with the speed of electron travel in the high
vacuum interelectrode space of a tube. Any abrupt change of the
external forcing function requires some finite time before it makes itself
felt as the collector. First carriers must be injected, and then they
have to travel across the junction. In turning off the transistors, cur
rent will continue flowing until the carriers, previously injected, are
swept out. The transistor on and off times are proportional to the spac
ing of the basecollector, baseemitter junctions. With the new diffusion
techniques of producing thin base films and extremely small junctions,
the transit time can be made quite short. Special switching tran
sistors having onoff times of less than 25 imisec are currently available,
and improved manufacturing techniques give promise of even further
reduction.
The transistor currentamplification factor falls off with increasing
frequency as
<*(/) =
i +;'(///.)
where f a is the a cutoff frequency. When we use the transistor in a
groundedemitter circuit, the transformation £ = a/(l — «) yields the
Sec. 99] plategrid and collectorbase multivibrators 279
expression for the /3 variation with frequency
0(/) =
l+j[///.U«)]
Thus the /3 cutoff frequency is /„(1 — a), and since a is only slightly less
than 1, //j is a very small percentage of /«. In poor transistors, the /?
frequency may even be as low as 5 kc, but in the better ones it rises to
100 or 200 kc. At some sufficiently high frequency the reduction in
will cause the loop gain to drop below unity. There can no longer be any
regenerative action above this point. If excessively narrow pulses are
used for switching, the loop amplification of their highfrequency com
ponents may be insufficient to ensure a change of circuit state. For
highspeed switching, we must always look for transistors having a high
a cutoff frequency.
Operation in the transistor saturation region is accompanied by the
injection of minority carriers into the base region from both the collector
and the emitter. When we try to turn off the transistor, forward current
continues flowing until these carriers are swept or diffused out. But this
takes an appreciable time, usually about ten times as long as the onoff
time of the transistor in its active region. Therefore the transistor must
never be allowed to saturate in highspeed switching circuits.
If we attempt to prevent transistor saturation by increasing the size of
the bias resistor, so that R > f}R c [Eq. (918)], then both the collector
drop and the unstable period become dependent on 0. In addition,
during the collector recovery, the charging current also flows through the
base emitter circuit of the on transistor. Unless the collector voltage
change is severely restricted to allow for the additional base current flow,
the transistor may be driven into saturation for a portion of the cycle.
Our object, then, is to prevent the transistor from bottoming. One
method which might be used is to connect a diode from the collector to a
small external bias voltage. Once the collector falls to this voltage, the
diode conducts and maintains the transistor in its active region. Only
diodes which have fast recovery time may be used; otherwise the storage
problem will simply be transferred from the transistor to the diode.
A second possible circuit configuration involves shunting the collector
load by a Zener diode (Fig. 930o). This diode fires when the drop
across R c equals E, and subsequently operates in a region where there
are no carrier storage delays. The load resistance is now shunted by
r„ reducing the loop gain well below the value needed to sustain regenera
tion. Figure 930b illustrates the abrupt change in the path of operation
once the diode fires. In addition, if the circuit is designed to saturate
in the absence of the Zener diode, then the change in the collector voltage
280
SWITCHING
[Chap. 9
is always E z and the sweep period is again independent of the transistor
parameters.
Many alternative methods of preventing transistor saturation are in
common use. They differ only in the circuit location of the diodes and
in their particular firing points.
Fig. 930. Circuit to prevent bottoming through the use of a Zener diode, (a)
Transistor collector circuit; (b) transistor characteristic and path of operation.
910. Multivibrator Triggering and Synchronization. The minimum
trigger amplitude depends on its point of insertion. Once this has been
decided, we can draw a model of the multivibrator and include the pulse
generator along with its source impedance. Since we know the loading
introduced and the changeover point of the circuit, the required pulse
amplitude is easily calculated.
As an example, consider the problem involved when we apply a nega
tive pulse to the grid of the on tube. It must be large enough so that the
amplified plate pulse, which is coupled to the grid of the off tube, can
turn the second tube on. We gain one stage of amplification, but the
trigger source is loaded by the low impedance of a conducting grid. On
the other hand, a positive pulse applied to the off grid will have abso
lutely no effect until it is large enough to bring the tube into the active
region. It will always see a very high impedance, that of a nonconduct
ing grid.
Exactly the same problems must be faced in triggering the transistor
multivibrator except that now the trigger source should be able to supply
the base current requirements when turning it on or off, as the case may
be.
Diodes are almost always used in injecting the trigger pulses, for two
reasons: first, to decouple the two bases or grids until the application of a
trigger and, second, to prevent a pulse of the wrong polarity from falsely
triggering the multivibrator. This might happen if a rectangular pulse
Sec. 910] plategbid and collectorbase multivibrators
281
were coupled to both grids through a small capacitor. It would be
differentiated, and the positive input trigger, produced from the leading
edge, would turn the off tube on. The trailing edge of the applied input
also supplies a trigger, a negative pulse which may now turn the on tube
off and thus return the circuit to its original state. Decoupling through
diodes would prevent the second pulse from ever appearing at a grid
and falsely retriggering the tube.
We may also inject a synchronizing signal into one or both bases (grids)
of the astable multivibrator and lock its freerunning period to some
submultiple of the sync frequency. The charging curve would no longer
be a simple exponential but would have the external signal superimposed.
The tube conducts when the total grid voltage reaches cutoff (Fig. 93 la).
And as was done in Chap. 5, the sync signal might be considered as
Fig. 931. Sinewave synchronization of a vacuumtube multivibrator.
effectively changing the tube cutoff voltage (Fig. 9316). If we approxi
mate the exponential sweep by a straight line, then all the results of
Chap. 5 dealing with synchronization may be applied to the astable
multivibrator. In Fig. 931, the broken lines indicate the normal free
running waveshapes, while the solid curves are the ones due to synchro
nization by the sinusoidal signal e„
PROBLEMS
91. Assuming that the amplifier of Fig. 91 is a perfect amplifier having a constant
gain A = 10, design a twodiode feedback network p, such that the system's stable
points are at e in = ±10 volts and the switching points are at ei„ = ±5 volts. You
may use any required resistances and voltage sources in addition to two diodes in
designing the passive network. Let the smallest resistor equal 10 K. Sketch the
transfer characteristics and the operating line.
92. (a) Draw the loop transfer characteristics for the active network of Fig. 932.
Assume that all amplifiers and feedback networks are ideal over the complete range.
What are the coordinates of the stable points, and how large must the input pulses
be for triggering?
(6) Repeat part o when 0i is changed from 0.1 to 0.01.
282
200
SWITCHING
M
[Chap. 9
A, — 10
/.
<>eo
/3 2 0.1
(3,0.1
50v= — 20v
"I T
Fig. 932
93. The bistable multivibrator of Fig. 93 employs a 12AU7 with equal plate
resistors of 20 K. The plate supply is 200 volts, and the coupling network consists
of Ri = 0.5 megohm and R* = 1 megohm. Evaluate the states of the two tubes
as a function of E„. Tabulate the results and state which conditions will give
the greatest output swing.
94. An asymmetrical bistable multivibrator uses tubes having /» = 25, r p =■ 10 K,
and r« = 1 K. One plate resistor is 40 K, and the other is 10 K. The coupling net
works are identical with Ri  R t  200 K. Ew, = 250 volts, and E„  200 volts.
(a) Sketch and label the waveshape at each plate when alternate positive and
negative pulses are injected into the grid of T\.
(b) What are the minimum amplitude pulses required to cause switching? To
which grid must they be applied?
95. Design a bistable multivibrator using a 12AX7 and two 200volt power sup
plies. The plate swing must be 125 volts as the tube switches from cutoff to full on.
Specify all resistors and show how this circuit would be triggered by a chain of negative
pulses.
96. By coupling the cathodes of the two tubes of a bistable multivibrator together,
as shown in Fig. 933, the need for a second power supply is eliminated. For this
circuit calculate the values of R K and Rl that are required to keep one tube just cut
off when the other tube just reaches saturation. What is the function of Ck, and
must it be very large or can it be relatively small?
+300 v
Fio. 933
97. During the transition from on to off, the behavior of the bistable multivibrator
may be approximately represented by the model of Fig. 934o. In this model C
includes the stray, interelectrode, and speedup capacity. By taking the TheVenin
PLATEGRID AND COLLECTORBASE MULTIVIBRATORS
283
equivalent across each capacitor, the model may be simplified to the twoloop net
work shown in Pig. 9346, where e n is the equivalent pulse source inserted for triggering.
(o) Find the poles of the equivalent network by writing the mesh equations as a
function of p, setting the network determinant equal to zero, and solving for p t and p s .
(6) Show that the form of the transient response of the networks is
e. = Ae"' + Be*'
(c) If ft = 100, C =■ 50 put, Rt — 10 K, Bi  500 K, and the impulse of voltage
applied is infinitesimally narrow but with an area of 10  ' voltsec, find the time
required for e„ to increase by 100 volts.
zc
R'z
R'l
1
I 1 "
cz
fcW
t
1
] +
~
T
1
(b)
Fig. 934
88. Sketch the waveshapes at the base of r, and the collector of T t from I =
for the circuit of Fig. 935. Initially, T t is cut off.
99. (o) Draw the transfer characteristics and the unity gain line for the multi
vibrator of Fig. 935. Evaluate the minimumsize trigger (current) necessary for
proper switching.
(6) Calculate the maximumsize resistivecoupling network necessary to ensure
reliable operation. Repeat part a for this condition.
+20v
*l
,
c
in
 _ ,_,
10
t, msec
1
2
3
4
h '
070
Fig. 935
284
SWITCHING
[Chap. 9
910. A transistor having & = 25 is to be used in the circuit of Fig. 99. If R„ = IK
and Eu = 2 volts, specify a value for B that will allow for a ± 50 per cent variation in
fl between transistors and will still permit reliable operation. Calculate the change
in collector voltage from state to state.
911. A simple directcoupled bistable multivibrator makes use of the small back
voltage which must be overcome before the transistor turns on. In addition to the
characteristic shown in Fig. 936, the collectortoemitter saturation resistance of the
transistor is 50 ohms and in the active region = 40.
(a) Show that the circuit of Fig. 936 is a perfectly stable circuit having two distinct
states.
(6) Derive the magnitude of the minimum switching pulse.
(c) Calculate the voltage change at the collector.
(d) To what value must the collector resistor be changed if we want to ensure that
the transistor is never driven into saturation? Would the circuit function properly
under this condition?
? + 1.0v
912. (a) Show at least three ways of ensuring the triggering of a bistable vacuum
tube multivibrator on each pulse. A train of positive pulses or a train of negative
pulses is all that is available.
(6) Repeat part a for the transistor multivibrators of Sec. 93.
(c) Discuss the behavior of a bistable if a narrow pulse is coupled into both active
elements simultaneously through a small capacitor. Consider both polarity pulses
injected into the control terminals (i.e., the base or the grid).
913. We wish to complete the design of the monostable circuit shown in Fig. 937.
(a) Calculate the value of R c needed to drive T t just to saturation in the stable state.
(6) What value of C is required to keep the circuit in its quasistable state for
1 msec?
/3 = 40
Fig. 937
PLATEGRID AND COLLECTORBASE MULTIVIBRATORS 285
(c) Sketch the waveforms of the base and collector voltages of both transistors
when a pulse is applied to the base of Ti at t = 0. Label all break points and time
constants numerically.
914. The transistor monostable of Fig. 938 is triggered at t = 0. Calculate the
waveshapes at the base and at the collector of T it giving all times and time constants.
916. In Fig. 938, the timing resistor R is returned to a variable supply rather than
to 10 volts. Plot the pulse duration as a function of this control voltage E for the
range of 1 to 25 volts.
+ 10v
Fig. 938
916. The monostable multivibrator of Fig. 938 is modified by connecting a Zener
diode which will fire at 6 volts from each collector to Em This prevents the transistors
from being driven into saturation and improves the recovery time (Sec. 99).
(a) Draw the resultant circuit.
(6) Repeat Prob. 914 for this new circuit.
(c) Plot the operating locus of the transistor on the collector characteristics.
917. The monostable multivibrator of Fig. 914 employs a 12AX7 adjusted so
that the grid of the off tube is normally at —30 volts. Calculate the required value of
Eec if the other components are Rl = 100 K, R — 1 megohm, Ri = 1 megohm,
Rz = 2 megohms, and E» = 250 volts. Find the required value of C to make the
output pulse duration 100 /usee. Sketch and label the waveshapes appearing at the
grid and plate of the normally off tube after a trigger is injected.
918. Design a monostable multivibrator that will generate a 150volt 2msec
pulse at the plate of the normally on tube. Use a 12AU7 returned to 250 volts,
E„ — — 150 volts, and a +20volt trigger pulse. Specify all component values and
show where and how the trigger is applied.
919. Consider the monostable circuit of Fig. 939, where the voltage at both plates
is limited by bottoming diodes.
(a) Sketch to scale the waveshapes appearing at the plate and grid of Tt after a
large positive trigger is momentarily applied to the grid of Ti.
(b) Repeat part o when the timing resistor jBi is returned to ground and adjusted
to produce the same duration pulse as found in part o. Specify the new value of Ri.
(c) The tube parameters may be expected to vary by +30 per cent from the nomi
nal values. In light of these expected variations from tube to tube, what function
do the diodes perform?
920. The monostable circuit of Fig. 939 is triggered by a pulse applied through a
diode to one grid. If the internal impedance of the trigger generator is 500 ohms,
calculate the minimumamplitude trigger (opencircuit) under the following conditions :
(a) A positive trigger is injected at the grid of T\.
(b) A negative pulse is applied to the grid of T t .
Be careful to check the state of each tube immediately after switching.
286
SWITCHING
?+250v
[Chap. 9
7.5 K
021. When the astable multivibrator of Fig. 919 is made symmetrical, it will
generate a square wave. Calculate and sketch the waveshapes at one tube if 2J» =■
300 volts, R L = 50 K, n = 70, r p = 20 K, r e = 500, R = 500 K, and C  0.001 „{.
922. Design an astable multivibrator using a 12AXJ7 and having a plate swing of
150 volts. Ti should be on for 100 /usee, and T t for 900 /usee. Use a plate supply of
250 volts. Sketch the plate and grid waveshapes, checking that each tube recovers
completely before reswitching.
923. We desire to build an astable multivibrator for use as a squarewave generator.
The output at the plates should be 200 volts peak to peak and should have a dc level
of 200 volts and a period of 10 msec. At each plate we insert a limiting diode to
remove any overshoot which would otherwise appear. If the tube available has
M = 70 and r„ = 50 K, specify all other circuit components.
924, Consider the application of the complete plate waveshape of Fig. 921 (broken
line) to the gridcircuit timing network of the astable multivibrator. This may be
represented as shown in Fig. 940. Assume that the capacitor is initially uncharged
and that the output voltage e c is zero at t = 0. The excitation ei is
e t = —150  50e"^ t 4  60 ^sec
The interval of interest to us is where e c < —10 volts.
(a) Calculate the exact output waveshape during the interval that the second tube
is maintained off. What is the length of this interval? Compare this result with the
approximate solution.
(6) The recovery time constant t< is increased until the time at which the next
tube turns on is changed by 5 per cent. What is the ratio of the excitation time con
stant t« to the circuit time constant under this condition? Is the approximation
made in the text valid?
0.001 »f 2M
9 t T — VW 1
T
♦
=300V
+— L
Fig. 940
926. Design an astable multivibrator that will generate a pulse whose duration is
onetenth that of the total period of 2 msec. Use transistors having /S = 30, a power
PLATEGBID AND COLLECTORBASE MTJLTIVIBBATOBS
287
supply of 10 volts, and two collector resistors of 500 ohms. Make allowance in your
design for a 20 per cent variation in 0; check to see if recovery is complete before the
circuit reswitches.
926. In a symmetrical transistor astable multivibrator (Fig. 922), the supply
voltage varies from 10 to 30 volts. R < pR c , where R is the base bias resistor and R e
is the collector load resistor. Calculate the variation in the period of the multivibrator
due to the powersupply variation. Explain your answer.
927. The symmetrical astable of Fig. 922 employs the following components:
Ea = 10 volts, Re = 1 K, R = 50 K, C  1 juf, and = 25. Sketch the base and
collector waveshapes of one transistor for a complete cycle, labeling them with all
voltage values, times, and time constants.
928. In the multivibrator of Prob. 927, R is reduced to 20 K, a value that ensures
transistor saturation. Moreover, to guarantee fast recovery, the collector resistance
is shunted by a Zener diode which will fire at 8 volts (Fig. 930).
(a) Plot the collector and base waveshapes, giving all times and time constants.
(b) Hot the voltampere characteristics of the collector load.
(c) How will the Zener diode influence the sweep time stability of this circuit?
929. The inductively timed monostable of Fig. 927 uses a coil having L = 2henrys
and Ri = 10 ohms as its basic timing element. As the collector load of the second
transistor we use a 2ohm resistor, with Ri = 90 ohms and R t = 200 ohms. The
power supply is 5 volts, and the switching transistor has /3 = 25 in its active region.
(o) Calculate the current and voltage waveshapes at the collector of Ti after the
injection of a trigger.
(6) Repeat part o when is reduced to 20.
(e) If the basetoemitter resistance of the saturated transistor is 2 ohms and the
collectortoemitter resistance is 5 ohms, how would the waveshapes of part a be
modified?
930. Figure 941 illustrates an alternative configuration for an inductively timed
multivibrator. Show that the normal state is with Ti on and T t cut off. Sketch
the collector and base waveshapes of T 2 and the current in the coil after a trigger is
applied. Give all times and time constants and the voltage and current coordinates
at the break points. What are the advantages or disadvantages of this multivibrator
compared with the circuit of Fig. 927?
1
+ 10v
J 50
;ioo
Ti
50 /T>
AAVMT
Irf
z
1,000
— w\
0=25
Fio. 941
931. (a) Derive Eq. (937).
(b) Show that the complete restriction on x must be
0.5 < x < 1.0
288 switching [Chap. 9
(c) For the case of ti = t s , what are the stable states of the two transistors?
(d) Under the conditions of part c, calculate the loop current gain of the circuit.
Explain the significance of your answer.
932. The symmetrical multivibrator of Fig. 922 is synchronized by injecting
positive pulses into both bases through diodes. Because of the loading by the on
transistor, we can assume that these pulses will affect only the off transistor and that
they will act to shorten the off time. For simplicity, we shall further assume that
the base charging curve is absolutely linear, taking ti sec to rise from — Ea to zero.
(a) Calculate and plot the regions of synchronization for n = 1 and n = 2.
(6) Repeat part o if the pulses are injected into only one base.
PIBLIOGRAPHY
Abraham, H., and E. Bloch: Le Multivibrateur, Ann. phys., vol. 12, p. 237, 1919.
Chance, B., et al.: "Waveforms," Massachusetts Institute of Technology Radiation
Laboratory Series, vol. 19, McGrawHill Book Company, Inc., New York, 1949.
Clarke, K. K., and M. V. Joyce: "Transistor Circuit Analysis," Addison Wesley
Publishing Company, Reading, Mass., in press.
Eccles, W. H., and F. W. Jordan: A Trigger Relay Utilizing Three Electrode Thermi
onic Vacuum Tubes, Radio Rev., vol. 1, no. 3, pp. 143146, 1919.
Feinberg, R.: Symmetrical and Asymmetrical Multivibrators, Wireless Eng., vol. 26,
pp. 153158, 326330, 1949.
Linvill, J. G.: Nonsaturating Pulse Circuits Using Two Junction Transistors, Proc.
IRE, vol. 43, no. 12, pp. 18261834, 1954.
Millman, J., and H. Taub: "Pulse and Digital Circuits," McGrawHill Book Com
pany, Inc., New York, 1956.
Puckle, O. S.: "Time Bases," 2d ed., John Wiley & Sons, Inc., New York, 1951.
Reintjes, J. F., and G. T. Coate: "Principles of Radar," 3d ed., McGrawHill Book
Company, Inc., New York, 1952.
Toomin, H. : Switching Action of the EcclesJordan Trigger Circuit, Rev. Sci. Instr.,
vol. 10, pp. 191192, June, 1939.
CHAPTER 10
EMITTERCOUPLED AND CATHODECOUPLED
MULTIVIBRATORS
101. Transistor Emittercoupled Multivibrator — Monostable Oper
ation. The emittercoupled multivibrator and the equivalent vacuum
tube circuit, the cathodecoupled multivibrator, are circuits worthy of
more than a cursory glance, illustrating asr they do the possibility of one
circuit operating in any one of several modes. Selection of the particular
mode and a measure of control over the performance within this mode are
a function of the control voltage E\, which is labeled in the emitter
coupled circuit of Fig. 101.
+30 v
i\x)R d
Ei^xEbb
Fig. 101. Emittercoupled multivibrator — component values shown for Examples
101 and 102.
Because a general treatment often obscures the detailed functioning of
the switching circuit, as an introduction, one operating point will be
explicitly stated and the resultant circuit analyzed. By doing this, a
straightforward study may be made of the time response without, at the
same time, being forced to account for the effects of a varying control
voltage. In Sec. 102 the possible modes will be denned and any required
modification of our original treatment may then be considered.
We shall assume that the circuit of Fig. 101 is adjusted so that, in its
normal state, Ti is full on and T x is cut off. The application of a positive
trigger at the base of T x switches the multivibrator into the single quasi
stable state: Ti is driven into its active region, and its collector voltage
drop, coupled through C, turns Tj off. Transistor T lf now operating
289
290
SWITCHING
[Chap. 10
as an emitter follower, stabilizes with an emittertoground voltage of
approximately E%. As C begins charging, the basetoground voltage
of T 2 rises toward Ebb Once it reaches the new emitter voltage and
Ci, e 2 equals zero, T 2 will turn back on. The increased current in R 3
raises the commonemitter voltage above E h and it follows that T\ must
be cut off. Ed jumps to a higher value, driving Tz back into saturation.
Note that the coupling from Ti to Ti is through both the common
emitter resistance R 3 and the collector base network of C and R. But
coupling from Ti back to Ti is only through the emitter resistor, which
leaves R 2 relatively isolated from the transmission path. It therefore
serves as a convenient place from which to take the output. At this point
a small amount of capacitive loading may be tolerated without seriously
affecting the pulse duration.
In order to simplify the quantitative analysis of the multivibrator of
Fig. 101, we shall make the following further assumptions:
1. j8 is very large, allowing R S> R x and R S> Ri.
2. The TheVenin equivalent impedance looking from the base of Ti
into the control voltage potentiometer is so very small that it also will be
neglected.
Preceding switching, the values of all voltages, measured with respect
to ground, are
e cl (0") = En (10la)
en(O) = xEn = Ei (1016)
R,
ew(0) = e c2 (0)
e 8 (0) = E* 
6.(0) =
Rs
Ri ■+ Rz
En, =
Ri + Rz
Rz
Eh
Rz \ its
En
(10lc)
(10ld)
where e 8 (0~) is the initial charge on the capacitor.
— fyVV— "°  ( o—WA —
«!
Ebb^
(a) (b)
Fig. 102. Emittercoupled multivibrator — models holding for the quasistable state.
(a) Circuit of Ti\ (6) TheVenin equivalent circuit coupling the collector of Ti to the
base of T 2 .
Immediately after switching, the models of Fig. 102 represent the
multivibrator's state and must be used for the calculation of the system's
time response. From Fig. 102o, we can see that the conditions at the
SEC. 101] EMITTERCOUPLED MULTIVIBBATOBS 291
base and emitter of T\, subsequent to its transition into the unstable
region, are
en(0+) = e„(0+) = xE» (102o)
*» (0+) = jfirm (10 " 2&)
Therefore its collector voltage becomes
e cX (0+) = E» faiRt
where the second term of Eq. (103) is simply the drop in collector voltage
upon switching. Because our original assumption (R 3> RJ allows us to
ignore the loading of R i by R, the voltage fall at the base of T 2 will be
identically the drop of Eq. (103). We may immediately write
Recovery from this sharp change of voltage is toward E^, with the time
constant
ti = (Ri + R)C £* RC
Once e it i = 0, or from Eq. (102a), once the basetoground voltage of
T t reaches
€b2(ii) = e, = xEm,
Ti turns back on. We now know the initial, the final, and the steady
state voltage at the base of T^; their substitution into, and the solution
of, the exponential response equation yields
Ell— ^ 3 + X ^A
\ i?2 + Rx Rz J
h=T ^ Wl*) " (10 ' 5)
for the pulse duration.
The multivibrator reenters the regenerative transition region with the
base of r 2 at a level lower than its steadystate voltage [Eq. (10lc)].
We suspect that there will be a voltage jump, with the probability that
this jump will be larger than necessary to return the base to its original
point. The new base, emitter, and collector voltages can easily be
calculated by drawing the model holding after the jump, i.e., during
recovery (Fig. 103).
Just before the multivibrator switches, the final voltage across C is
e.«i) = E. t  xEn = Ev, \\  x(l + fAl (106)
292 switching [Chap. 10
By substituting this value into Fig. 103, we arrive at the new operating
point of the transistor. The algebra is much simpler when we can use
numbers, and the solution will therefore be left to the reader as a part
Fig. 103. Recovery model of the emittercoupled multivibrator.
20.5
Fig. 104. Emittercoupled multivibrator waveshapes — monostable operation — mode
12. Values given are for the solution of Example 101.
of his analysis of specific problems and to the calculations performed in
Example 101.
After the final jump, the complete circuit recovers toward the original
steadystate conditions, with the time constant
r, = C(i?i + Ri  R 3 )
Sec. 102] emittercoupled mtiltivibratobs 293
Waveshapes appear in Fig. 104 with all jumps and time constants indi
cated. Observe that the base, emitter, and collector of T 2 all have
identical recovery waveshapes (the saturated transistor is a short circuit).
Example 101. By setting x = 0.4, the monostable multivibrator of Fig. 101 is
adjusted to operate in the mode described above, where T t switches between its
cutoff and active regions. All circuit components are as given in Fig. 101. We do
not need to specify C for the following analysis, since it would be selected on the basis
of the required pulse width.
The duration of the unstable state, which is of primary interest, is controlled by
the charge in C, maintaining T, cut off. Only two unknown voltages must be found
in order to evaluate this interval, the initial value at the base of T t as the circuit enters
its unstable region and the voltage at which T% again turns on.
From Eq. (10lc) the stable base voltage of T t is
800
e M (0) = jgQ 30 volts = 18.5 volts
With x = 0.4, the voltage at the base of Ti remains fixed at
en = En = 12 volts
When the trigger is injected, Ti turns on and the commonemitter voltage drops from
18.5 to 12 volts {xE»).
If the small base current flow is neglected, the voltage drop across ffii will be propor
tional to the 12volt drop across R%:
jRi _ AEg
Rl xEhb
or AE el = 15 volts. C couples this change to base of Tj, driving it from its original
value of 18.5 down to 3.5 volts. It immediately starts charging, as shown in Fig.
1026, toward 30 volts, with the time constant n. When eui = or when en =
12 volts, Tt turns back on. Consequently
, ,„ 30  3.5
ti = ti In _ = O.dSSri
At the end of the sweep interval the voltage across C is only 3 volts. Substituting
this value into the model of Fig. 103, we find that the base of Ti jumps from 12 to
20.5 volts. With the time constant
tj  1.310C
the circuit returns to the original conditions. These voltages are used to label the
typical circuit waveshapes of Fig. 104.
102. Modes of Operation of the Emittercoupled Multivibrator. The
commonelement (emitter) coupling permits this multivibrator to oper
ate in a possible multiplicity of modes in each of its two classes. Since
Ti is always conducting in the normal state, these classes might well be
defined in terms of the quiescent condition of this transistor: class 1
existing when the bias current is adjusted so that T% is placed in its
294
SWITCHING
[Chap. 10
active region, and class 2 when it is saturated. The particular class
depends solely on the relationship of R to Ri; for T* saturated (class 2)
R < fiRz
We can prove the sufficiency of this condition by considering the transis
tor model holding under saturation [also see the derivation of Eq. (918)].
Class 2 operation is to be preferred over class 1 ; here both the initial
conditions and sweep duration are relatively independent of the transistor
parameters. However, the saturated transistor has a long recovery
time and we would be forced to incorporate antibottoming diodes. We
thus transfer the minoritycarrierstorage problem from the transistor
to the diodes, where it is more readily handled.
The various possible modes, within each class, are determined by the
switching conditions of T\, i.e., its starting and ending states. Modes
will be classified by two numbers, the first describing the state of T\, as
listed below, and the second the class (state of r 2 ).
States of T\
Before switching After switching
Cutoff
Cut off
Cutoff
Part on
Full on
Cutoff
Part on
Full on
Full on
Full on
Since the state of T\ is a direct function of E\, investigation of the
multivibrator's performance will proceed, following the potentiometer
setting, from x = to x = 1. Each mode will be delineated by establish
ing the boundary values (of x) in terms of the circuit parameters. At any
particular setting of x, detailed timeresponse calculations would simply
follow the analysis outlined in Sec. 101.
We shall mainly concern ourselves with modes n2, representing as
they do the more stable operation of the emittercoupled multivibrator.
In order to complete the discussion one astable mode of group n\ is
briefly treated in Sec. 104. All approximations and assumptions made
in Sec. 101 are also applicable in the following arguments.
Mode 02. In this mode Ti is normally off. Application of a trigger
turns T\ part on, but not far enough on so that its collector voltage change
can drive Tt into cutoff. Upon removal of the trigger, T 2 is still in the
active region, and it immediately switches back to its normal state, full
on. The output pulse generated is of small amplitude and of exactly the
same duration as the input trigger.
Sec. 102] emittercoupled multivibrators 295
By assuming that T t has turned full off and that Tx is still part on, the
model drawn is much simpler than the one required if both transistors are
taken in their active regions (the model for Tx is given in Fig. 102a).
The region's boundary remains the same regardless of the direction from
which we approach it. If Tz is cut off, the initial value of its base voltage
is given by Eq. (104). Whenever Tx is on, the drop across the emitter
resistor is always the control voltage xE^,. In order for T 2 to be part on,
* = (ot  '£)*>  xE » > ° (10  7)
Solving Eq. (107) for the limit of x, which satisfies the inequality stated,
we find that the limits of mode 02 are
°<*<OTm (10  8)
This calculation serves mainly to define the lower limit of mode 12, the
first useful range.
Mode 12. An input trigger will again switch Tx part on, but now the
drop in its collector voltage is suffi
cient to drive T 2 into cutoff. The
upper limit of x is at that particular "'4> l
potentiometer setting that would o—Tr E —
allow Tx to switch from cutoff to I = p I ble 1 bb "■"
saturation upon being triggered. =xE,
To find this setting, we should draw
jxE bb B 3 j09+Ui (
the model holding for Tx full on _, ,„,»,,,. tu v„* r
, . Fig. 105. Model for the upper limit of
(Fig. 105) and solve for the corre mo de 12.
sponding emittertoground voltage.
But for 8 » 1, 8 + 1 = /3, and therefore
e. = xE» S p ?' p E» (!0 9 )
tix T «3
The limits of x for operation in mode 12 are from the upper limit of
mode 02 to the value found from Eq. (109) :
Rl E * < x < D R * p (1010)
Ri + R s R* + Rs^ Rx + Ri
For the multivibrator of Example 101 the limits are 0.273 < x < 0.445.
Mode 12 was the one treated in detail in Sec. 101. Its pulse width,
defined by Eq. (105), is a function of the potentiometer setting x.
Increasing the setting increases both the drop in the base voltage of Tj
296 switching [Chap. 10
and the value of the emitter voltage after switching into the unstable
state. As a consequence of the greater separation between the initial and
final values of the base voltage, the pulse duration increases with potenti
ometer rotation. Substituting the limits of x into Eq. (105) gives the
permissible range of output pulse width as
°< fe <4 1 + B$7Ti£] (1(M1)
When the circuit is symmetrical (Ri — R 2 ), the maximum pulse width
generated is 2i tm « = n In 2.
Mode 22. In this mode T x switches from cutoff to full on. The
next possible circuit change occurs when the normal state of Ti changes
from cutoff to part on. But with T x cut off, Eqs. (1016) and (10lc)
hold: by solving for e he \ = 0, we find the upper limit of mode 22.
etei = xEn  ^^ ^* E» < (1012)
The limits of x for this mode become
■Rs R
R 1 + R* <X< RTTR, (1(M3)
and in Example 101, the upper limit is located at 0.615.
Note that unless Ri > R 2 , mode 22 will not exist and the transition will
be directly from mode 12 to 32. Furthermore, if R x = R 2 , the range of
voltage settings for mode 22 operation degenerates into a single point.
When Ti goes full on, the drop across R 3 is xEu,. The remaining
voltage must appear across R x ; therefore the drop in the collector volt
age of Ti will now be
Ae„i = (1  x)E» (1014)
This same drop appears at the base of T 2 , turning it off. The base
charges from its initial voltage, found from Eqs. (10lc) and (1014),
«m(0+) = jjpjrij; E»(l x)E vb (1015)
toward Ett, with the time constant n. It finally conducts when
e«(<2) = xEu,
Consequently, in mode 22, the pulse width becomes
h = r t In 2 ~ Ri/ ^l\ Ri) ~ X (1016)
Sec. 102] emittercoupled multivibkatoks 297
Mode 32. Under the circuit conditions existing within this mode of
operation, Ti always conducts, maintaining the emitter voltage fixed
at xEn, across the complete transition region. Limits, on x, are from
the upper limit of mode 22 to that
value which puts both transistors i 1
into permanent saturation (Fig. flj s Rl S
106). f f
The upper limit, found from the _[] X £».=■
model of Fig. 106, is =xE bb <# 3
R
xEu < ■= — ,, D 8 — 5 En, Fiq. 106. Both transistors in saturation.
Ri II «2 + R%
Thus, in mode 32, the range of x becomes
Ri ^  R% , 1 _i
Rz + R* <X< Rx  R* + R, {1 ° 17)
When R 3 is very large compared with Ri and R 2 , the limits of x for which
this mode holds shrink almost to the vanishing point. The circuit
conditions before switching are determined from the appropriate model
(Ti part on and T 2 saturated) :
en(O) = e» 2 (0) = e.(O) = xEn (1018a)
«.i(0) = E»  E» [^  (1 ~ R f Rl ] (10186)
But after switching,
eci(0+) = xEu, (1019)
The net change in the collector voltage of 7\ [Eqs. (10186) and (1019)]
is coupled to the base of T^, and as the multivibrator enters the quasi
stable region, Ae c i determines the initial value of e b2 :
e*m = (2x  1)E» + [^  (1 ~ R f Rl ] V* (1020)
As in all the other modes, the base charges toward En, switching at xEu,.
The pulse duration, still a function of x, turns out to be
t. = ri In *' \_* R *> (10.21)
In the three modes of operation that generate a usable pulse, its dura
tion, at least to the first approximation, is independent of £», resulting
in extremely good pulse stability.
298 switching [Chap. 10
Mode 42. This mode exists when both transistors are always satu
rated. Thus an input trigger will have absolutely no effect. The limits
are
Rz
Ri II Ri + Rz
< x < 1
(1022)
It is included only to complete the range of setting of the potentiometer
and, as mode 02, is of no practical use.
A summary of the results is presented in Tables 101 and 102. Table
101 shows the operating regions. Within each mode the normalized
pulse duration is given by
b + ax
tn
T 1 — X
(1023)
The constants a and b are listed in Table 102.
Table 101. Transistor States, Emittercoupled Multivibrator
Mode
Upper limit of x
T l
T 2
02
Rt Rs
Off
Full on
Ri 4 Rt R2 + Rt
12
R,
Off
I
part on
Full on
i
cut off
Ri + Rt
22
R,
Off
1
full on
Full on
1
cut off
R% 4~ Rz
32
R,
Part on
1
full on
Full on
I
cut off
Ri 1! ^2 + Rz
42
l
Full on
Full on
To summarize further the behavior of the multivibrator in terms of
the particular modes, the complete normalized period is plotted as a
function of x for the circuit of Example 101. The solid curve 1 of
Fig. 107 illustrates the varying pulse duration as a function of x when
the impedance of the control potentiometer is negligible. This cor
responds to the mode boundaries discussed above. Both the demarca
tion between the individual modes and the different rates of pulsewidth
variation with x within each mode are quite clear. It might be noted
that, in this special case, the pulse duration appears to vary almost
linearly with x within mode 12.
Sec. 102]
EMITTERCOUPLED MTJLTIVIBRATOBS
299
Table 102. Pulse Duration or Emittercoupled Multivibrator —
Modes n2
Mode
6*
a*
12
1 R '
Ri
Rz "h R3
R,
22
R>
1
Rt f R$
32
Ri
2+ R,
\ Rt Rt)
* Constants for Eq. (1023).
Effects of Finite Potentiometer Impedance. When the control potentiom
eter is of appreciable size, we must include its equivalent source imped
ance in all calculations involving the actual base and emitter voltage of Ti.
The dashed curve 2 of Fig. 107 shows the effect of a 10K potentiometer
0.8
0.7
0.6
i 0.5
!s 0.4
T
0.3
0.2
0.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 L0
Fig. 107. Pulse duration as a function of potentiometer setting.
on the pulse duration in the various modes of operation. This plot
is the result of inserting the Thevenin equivalent impedance of
R. = x(l  x)R d (1024)
in series with xEn.
For mode 12, the input model is shown in Fig. 108a. Here, for any
setting of x, the source impedance is much less than the activeregion
input impedance (/3 + 1)R 3 of the transistor. With the specific circuit
values of Example 101, the input impedance is 56 K compared with the
^
fl 2 500
#3=800
/370
B25K
 £,,6=30 v
Mode 22
1
*•
"?
\
.(
) _^
/
\
loc
n v j
'/
\
/
■/,
T
_Mode 12.
J
II
ft
1
It
II
e<
_J
II
L
300 switching [Chap. 10
maximum R a value of 2.5 K. The loading of the potentiometer is
negligible, and the base voltage is but slightly less than the opencircuit
voltage xEw,. To generate a givenduration pulse, the value of x must be
set slightly higher than previously found, in order to compensate for the
voltage drop across R,.
Once the multivibrator begins operating in mode 22, the input model
must be changed to the one shown in Fig. 1086, which is obtained
by replacing the transistor by the Thevenin equivalent circuit seen
looking into the saturated base. The input impedance is reduced
from 56 K to the parallel combination of Ri and R 3 (445 ohms). Fur
thermore, the potentiometer is varying about the midpoint of its
setting, and its impedance remains close to the maximum possible value
{R, = 2,500 ohms). Because most of the voltage will be developed
across R„ any change in the controlvoltage setting will have but small
x(Xx)r d x(lx)r d Cj R^Ri
J_ i»i t e i 445 _j_
(b)
Fig. 108. Equivalent input circuit of T u including the potentiometer source imped
ance, (a) Mode 12 operation; (6) Ti saturated (mode 22).
effect on the base voltage of TV The commonemitter voltage will not
vary much, and neither will the pulse duration. This segment of the
curve in Fig. 107 becomes much more horizontal. In Example 102
we shall consider one specific setting of x, lying in this mode, to illustrate
the role of R„ in determining the circuit voltages and timing.
Mode 32 operation is restricted to such a small fraction of the range
that, in general, it is of little interest. R, would be treated as above, by
including it in the appropriate models. The reader may perform this
substitution and calculation himself.
Example 102. The multivibrator of Fig. 101 uses a 10K potentiometer for its
timing control. When it is set at x = 0.55 the circuit is operating near the center of
mode 22. All other circuit components are the same as in Example 101, and the
results of that problem may be used in this solution.
The initial voltage at the base of T 2 , before triggering, is 18.5 volts (from Example
101). After the trigger turns 7\ on, the saturated transistor voltage may be found
from the model of Fig. 1086. Here xEu, = 16.5 volts, and R, = 2,475 ohms. From
Fig. 1086,
eei »,«. 13.33 + ^~ X (16.5  13.33)
= 13.81 volts
If R, were 0, this voltage would be xEu, = 16.5 volts. The loading of the potentiom
eter reduces the transistor voltage to slightly above its saturation value. Thus, as 7\
Sec. 103] emittercoupled mtjltivibbatoks 301
turns on, the drop at its colleetor is only
A« el = 30  13.81 = 16.2 volts
This drop is coupled by C, driving the base of Tt down to
en(0+) = 18.5  16.2 = 2.3 volts
Recovery is toward 30 volts, with the circuit reswitching at the commonemitter
voltage of 13.8 volts:
When the source impedance is small enough to be neglected, the voltage at the
terminals of 7\ will be xEu (16.5 volts). In this case, the change in collector voltage
is only 13.5 volts and the base of T% is driven to 5 volts. The reswitching value is also
higher, 16.5 volts, and since a greater percentage of the total exponential is used, the
sweep period becomes somewhat longer:
*' Tim 30°~16.5 =  615r '
103. Monostable Pulse Variation. The question confronting us in
this section is, What circuit relationships must be satisfied to achieve
some physically permissible variation of the pulse duration with respect
to x? In modes 12 and 32, the function might well be a linear one,
and in mode 22 it might be either constant or linear. One commonly
used method of attacking this problem is first to expand the general time
equation (1023) into some series and then, by examining the individual
terms, attempt to eliminate those that contribute the nonlinearity.
But since x takes on a relatively large range of values, there is no possible
series that will converge fast enough so that all the higherorder terms
in x can be eliminated. A solution of this type is quite tedious, and in
addition, the design requirements are often obscured beneath the algebra
needed in handling the existent higherorder terms.
A second, more fruitful approach is through an investigation of the
properties of the slope of the curve rather than of the curve itself. When
we differentiate Eq. (1023) with respect to x, the resultant expression
for the slope is no longer an exponential function of x, and consequently
it is of simpler form than the original equation :
S = M . (» + .) (10 _ 25)
ax ax 2 — (a — b)x — b v
In order for the pulsewidth variation with respect to changes in x to be
linear, the slope must be independent of a;; it should be a constant. But
this requires that the denominator of Eq. (1025),
y = ox 2  (a  b)x  b (1026)
302 switching [Chap. 10
must also be a constant for all values of x within any one mode. Equa
tion (1026) is the equation of a parabola, and therefore it is impossible
to satisfy the required condition; we can never expect a completely linear
relationship. A good compromise will keep the change in y small for
the range of x within the mode under investigation.
We would conclude, from scrutinizing the properties of a parabola,
that it is most nearly constant with respect to small variations of x in
the vicinity of its vertex. The best design would adjust the circuit
parameters so that the point of inflection of Eq. (1026) lies in the middle
of the mode in which the linear function is desired. Differentiating
Eq. (1026) and equating to zero locates the vertex at
For mode 12 of the multivibrator that was treated in making the plot of
Fig. 107 (zero controlsource impedance), x mi is at 0.345 compared with
the lower and higher limits of Xn = 0.272 and Xm = 0.444. It is quite
close to the optimum placement of the vertex at x = 0.358. But in
mode 22 of the same circuit x m % is at 1.19, well outside the mode limits of
xn = 0.444 and Xk% = 0.614. The difference in linearity is quite apparent.
The second condition for good linearity also comes from the examina
tion of the properties of the parabola. If the focus is close to the vertex,
then the parabola is wide open and will be more nearly constant in the
vicinity of the vertex, further improving the linearity. This distance,
for the parabola of Eq. (1026), is
4a
Generally it is not possible to make D small and still satisfy Eq. (1027).
Setting D = in mode 12 is a trivial case, requiring a zero value for R it
which would make the multivibrator inoperative. But even in this case,
setting R 3 as small as possible compared with Ri minimizes D within the
other design constraints that now determine the lower limit of .fi^ The
more important condition is the one given by Eq. (1027), and this is the
one which must be satisfied for best linearity.
Under some other circumstances it may be desirable to make the pulse
width relatively independent of x. From Eq. (1025) it can be seen that
this requires a = — b. All modes give ridiculous answers in satisfying
this condition, either a pulse duration of zero or negative values of circuit
resistors.
By including the source impedance of the control path in the deriva
tion of the timing waveshape, we would find it possible to maintain the
pulse width almost constant with respect to small changes in a; in mode 22.
Sec. 104]
EMITTERCOUPLED MULTIVIBRATORS
303
The equation resulting would be in the same form as Eq. (1023), with,
of course, different constants. Either from this equation or by consider
ing the effect on the base voltage of Tx (as in Example 102), the slope
of the timeversusx curve can be minimized by maximizing R,. Any
changes in the transistor parameters would now have but little influence
on the timing, and the sweep stability would be greatly improved.
104. Emittercoupled Astable Multivibrator. Astable operation, at
least according to Sec. 97, apparently is dependent upon two independent
energystorage elements maintaining first one and then the other tran
sistor cut off. But in the circuit configuration of Fig. 101 or 109, a
single capacitor controls the duration of both unstable states. To see
how, suppose that, with the feedback loop broken and by satisfying the
following conditions, we bias both transistors in their active region.
For r 2 active:
R > fiRz
For Tt active:
(1 + g)fl 8
R + Q.+ fi)R,
< x <
R 3 R + (1 + fi)RzRi
R(Rt + R 3 ) + (1 + /»)«,«!
(1028)
+25 v
The complete limits of astable operation expressed by Eq. (1028)
are readily derived from the appropriate models, one with Ti cut off and
5" 2 active, and the other for Ti saturated while T 2 remains active.
Once the feedback loop is again closed, the regenerative action of the
multivibrator rapidly drives T 2 into cutoff. As C recovers toward 2J»,
the base of T 3 eventually reaches the
emitter voltage xEu,, at which point
T2 reenters its active region. The
increased current through R t now
raises e, above xEu, and thus drives
Ti off. The resultant jump in the
collector voltage of Ti forces T 2 to
ward, or even into, full conduction,
capacitor charging current contrib
uting the necessary additional base
current flow. However, C will again
recover. In the process, the total
emitter current drops; the base voltage of T 2 and the commonemitter
voltage exponentially decay, finally recrossing the xEu line. At this
instant T t turns back on, and the cycle repeats. We see that the single
capacitor performs both timing functions; first it maintains T 2 cut off, and
next it supplies the additional current that keeps T 2 turned on and/or
Ti cut off.
Fro. 109. An astable emittercoupled
multivibrator.
304
SWITCHING
[Chap. 10
As a convenient starting point for the purpose of computation, we
might well look at the conditions of Fig. 109 while T 2 is on, just going off,
and Ti is off, on the verge of turning on. The initial voltages at t = Qr,
neglecting any charging current which might be flowing, are
e c i = E» — 25 volts
e&i = e& 2 = e, = xE bb = 7.5 volts
Gq — €el
e&2 = 175 volts
Immediately after the circuit changes state, at t = + , T 2 becomes cut
off and Ti active. Therefore the model with which we must be concerned
is the one shown in Fig. 1010a. Since the 100K base bias resistor of
!T 2 is so very much larger than the 1K collector resistor of T\, we may
?+25
•IK
100K
333
*3(l + /3)<
=■25 K<
(c)
Fig. 1010. Transistor models used for the calculation of astable operation, (a) 7\
on, T 2 off; (6) T^ off, T 2 saturated; (c) Ti off, T 2 active.
ignore the additional loading introduced by R. From Fig. 1010a, the
conditions at t = 0+ are
en = e, = 7.5 volts
7.5 volts
%b\ =
= 0.3 ma
(50 + 1)500
e cl = E»  504i (1 K) = 25
15 = 10 volts
The collector voltage falls 15 volts, and the base of T 2 drops by the same
amount, from 7.5 to —7.5 volts. It immediately starts charging toward
En with the time constant
tx = RC = 20 msec
But Tz turns back on once e 62 = e. = 7.5 volts, and thus the first portion
of the cycle lasts for
(l = Tl ln »J=5* = 20 In 2b ~ ( ~ 7 ; 5) = 12.36 msec
En
E,
25  7.5
Sec. 104] emittercoupled multivibrators 305
The final charge in C, which we shall need as a boundary condition in the
next operating region, is
e«('i) = e«si(<i) — e t2 (<i) = 2.5 volts
After Ti turns back on, the jump in the collector voltage of T x as it
turns off, coupled through C, forces T 2 into saturation (Fig. 10106).
Before we can calculate the time response, we must verify the correct
ness of this assumption. From the model drawn for the saturated tran
sistor, we see that the equivalentemitter voltage is
ElbRz 8.33 volts
R\ + #3
The additional capacitor charging current flowing through the 333ohm
source impedance (R*  Ri) raises the voltage of all the elements of T t
to 11.87 volts, a value well above the saturationthreshold voltage. The
initial current flow through C and the base of T» becomes
. ». 25  2.5  8.33 , na
»m(U) = ■ = 10.6 ma
However, the base current necessary to sustain saturation is merely
*'  (/J + l)fi, = 25K = ° 333 ma
and of this, the amount contributed by the normal bias current flow
through R is
Eg  e, 258.33
*» = — r  = ioo k = 0167 ma
Thus the equation of the base current component due to the capacitor
charging current becomes
iuc(t) = 10.6e""
where T2 = (fl t + R 2 1 R 3 )C = 0.266 msec
T% finally enters its active region (from saturation) when
ibtcih) = 0.167 ma
As a consequence of the large initial and small final value of current flow,
the duration of the first portion of the recovery period takes virtually
the complete exponential:
1 ft R
h = Tt In Q^Tjy = 4r 2 = 1.06 msec
At the end of this interval, a new model is needed for r 2 , one represent
ing the active region (Fig. 1010c). The transistor input resistance,
306
SWITCHING
[Chap. 10
measured from base of 7% to ground, is (0 + 1)22, = 25 K, and the
equivalent input opencircuit voltage is
(0 + 1)« 3 „ 25 K
En —
R + G8 + 1)«;
•Em, =
100 K + 25 K
25 volts = 5 volts
Moreover, since the transition from the saturation to the active region
will be smooth (without any voltage jump), there is no need to calculate
the initial charge on C. Charging continues, but now toward 5 volts,
with a new time constant t 3 = [(j3 + 1)R 3 \\ R]C = 6.7 msec. Once the
commonemitter voltage falls to 7.5 volts, Ti again conducts, the circuit
reswitches, and the cycle repeats:
U = t 3 In
5  8.33
5  7.5
= 1.94 msec
During this final interval Tt is in its active rather than its saturated
state. As its base current continues to decrease, the collector voltage
now starts rising. The generated waveshapes at the base of T 3 and at
the commonemitter junction appear in Fig. 1011.
Fig. 1011. Controlling waveshapes of emittercoupled astable multivibrator.
Sec. 105]
EMITTERCOUPLED MULTIVIBRATORS
307
106. Cathodecoupled Monostable Multivibrator. A vacuumtube
cathodecoupled circuit (Fig. 1012) functions in a manner quite similar
to that of the commonemitter transistor multivibrator; the principal
differences involve the circumstances that surround the change of circuit
state. These naturally arise from
the physical properties of the vac
uum tube and will be represented
in the circuit models. The various
operating modes would be defined
for the same switching conditions
as in the transistor circuit.
All aspects of this multivibrator's
operation in mode 12, the only one
with which we shall be concerned,
are presented in the two models
drawn below. The first represents
the circuit conditions both during
Fiq. 1012. Cathodecoupled monostable
multivibrator.
recovery and while the multivibrator is in its normal state (Fig. 1013a),
and the second, when it is in its quasistable state (Fig. 10136).
In an effort to simplify computations, we shall make the following
assumptions:
1. The equivalent grid resistance in the positive grid region, r„ is
small compared with R.
2. R is sufficiently large so that its loading of Ri may be neglected.
3. The contribution of grid current in developing the voltage across
Ri is insignificant.
From Fig. 1013a (T t off, T % saturated), the quiescent circuit voltages
are ,
e 5t 2(0) S
e ci (0~) S e*(0)
#3
R% + #3 + r p
r P + R 3
Eu
e g (0~) =• Em e c2 (0~) =
Eu
r P + R*
R2 + R% + T t
Eh,
(1029o)
(10296)
(1029c)
(1029d)
For proper operation in this mode, T\ must be cut off. The required
minimum bias voltage (grid to cathode) is determined by the plateto
cathode drop.
Eta\ — —
Ey,
En
et
(1030)
308 switching [Chap. 10
and to ensure that T\ remains off in its normal state,
»tki = xE»  et <  Ebb ~ e " (1031)
where e k — RzEti,/(rp + R 2 + R 3 ).
As the multivibrator enters upon its quasistable state, after the appli
cation of an external trigger, Ti switches from cutoff to its active region.
The plate and cathode voltages become (Fig. 10136)
e*(0+) =
r P + Ri+ (m + 1)R,
«n(0+) = E»
r P + Ri+ <jt + 1)B,
(1032)
(1033)
We must check that e*(0+) > xE^, because if this inequality is not
satisfied, the model of Fig. 10136 is no longer valid. It must then be
Fig. 1013. Cathodecoupled multivibrator circuit models, (o) Normal state; (6)
quasistable state.
replaced by the model representing positive grid operation. The second
term of Eq. (1033), the drop in plate voltage upon switching (Ae»i),
is coupled to the grid of Tt, driving it well below cutoff:
«c S (0+) = e c2 (0)  he n
RtEvb
Rx(\ + ;js)Ew
Ri + Ri + r p r P + Ri + (n + 1)2?,
(1034)
The grid of T t charges from its initial value [Eq. (1034)] toward En,
finally switching at t — h as it reaches its particular cutoff voltage:
e„*(ii) = e*(0+) 
En,  e>,(0+)
(1035)
Sec. 105] emittercoupled multivibrators 309
where e*(0+) was given in Eq. (1032). And for high/* tubes, where
( M + 1)R, » r, +• R u
The charge time constant, again found from Fig. 1013&, is
r, = C{« + R t  [r, + ie 3 (M + 1)]} £< «C
and thus the pulse duration becomes
"""ttSB < 10  36 >
If we substitute the two values of e c2 found above into Eq. (1036),
we shall see that the pulse width is independent of E a and, in addition,
that the resultant equation is of the same form as the general timing
equation (1023). For the particular case of a highju tube, we obtain the
equation
i R* , 1 + nx Ri
*inln Ri + R \+_ r ' x *— *« (1037)
At the end of the unstable state, the voltage across C is
e,(*i) = e»i(0+)  ea(h)
and by substituting this value into the circuit of Fig. 1013a, we can
compute the initial value of the positive grid excursion as the multi
vibrator starts recovering toward its normal state. Recovery at the
plate of Ti and the grid of T 2 is with the time constant
T 2 = C[fi, + r./(l  A k )]
where the positive gain from grid to cathode effectively multiplies the
grid resistance (A k < 1). We see, at both the plate and cathode of T 2 , the
reflected grid recovery waveshape (Fig. 1014), amplified by the appropri
ate factor. Waveshapes of all tube elements, further illustrating the
performance of this circuit, appear in Fig. 1014.
Limits of the various operating modes of the vacuumtube multi
vibrator are computed from the appropriate models by simply solving
for their boundary values. Since, in its active region, the vacuum tube
operates within a grid base, dependent not only upon the voltage across
the tube but also on the tube parameters, the factors defining the limits
become quite unwieldy. In the special case of the high/i tube, there
will be considerable simplification. However, for any specific multi
vibrator, where subcalculations using the circuit resistance and the tube
310
SWITCHING
[Chap. 10
parameters are usually performed, the problem is straightforward and
not at all difficult.
106. Limitation of Analysis. Physical multivibrators differ in several
important aspects from the idealized circuits treated in this chapter.
Experimentally measured results cannot be expected to agree exactly
with those calculated on the basis of our previous discussion. Since
we now understand the multivibrator's operation, we might take a
t, t
Fig. 1014. Cathodecoupled multivibrator waveshapes — monostable operation.
second look at the terms previously ignored, with a view toward finding
out how they would modify the original analysis.
One of the two terms affecting the pulse duration of the transistor
multivibrator is the small voltage drop, about 0.1 or 0.2 volt, that
actually appears between the base and emitter of the conducting tran
sistor. This voltage could be included in the transistor models by insert
ing a small battery in the base emitter circuit. If the transistor switches
to well below cutoff and starts charging toward E», it seems completely
reasonable to neglect 0.1 volt compared with the total base voltage change
Sec. 106] emittercoupled multivibrators 311
of 5 to 20 volts or more. However, near the lower limit of mode 12 and
the upper limit of mode 32, the transistor is driven only slightly below
cutoff and the resultant error in the pulse duration will be quite large.
The generated pulse will always be shorter than the calculated pulse, thus
introducing additional curvature in these regions of Fig. 107.
The second factor, which also tends to reduce the pulse duration, is
the temperaturedependent /„<>. This, the reverse collectortobase
current, contributes an additional charging component to C while T%
remains cut off. And of course faster charging means that the pulse
will be shorter than expected. I c0 may be 5 /ta or less at 20 9 C, but it
increases exponentially with temperature, even reaching 50 to 100 n&
at the elevated ambient temperatures often encountered. As long as the
normal charging current through R remains much larger than I c0 , the
temperature effects are minimized. Thus another reason appears for not
operating under modes nl, where R is, of necessity, quite large. Further
more, some form of temperature compensation is usually employed in
transistor circuits.
The extreme curvature of the plate characteristics at low values of
plate current leads to the tube's turning on at a lower grid voltage than
that given by the piecewiselinear model. Consequently the charging
exponential is interrupted sooner and the generated pulse will always be
shorter than calculated. A larger plate drop is necessary to ensure driv
ing the other tube into cutoff, and therefore the lower limit of mode 12
will occur at a higher setting of x and the upper limit of mode 32 at a
lower setting of x. The difference from the ideal would be appreciable
for 1owm tubes but would become negligible when high/* tubes are
employed.
The secondorder effects are primarily discussed to show that where
physical and theoretical results disagree, a closer look at our original
assumptions, or at the active elements themselves, will often explain the
source of discrepancy. Practical circuits always have adjustments for
timing. Since 5 or 10 per cent accurate components and 20 to 50 per cent
tolerance in tube and transistor parameters are what the designer must
cope with, exact design is neither possible nor desirable. Thus a rapid,
simple, approximate treatment is often more satisfactory than an exact
analysis, provided that the answers obtained are reasonable.
PROBLEMS
101. The emittercoupled circuit of Fig. 101 employs the following components:
fl,  R t = 2 K, R, = 1 K, R = 50 K, C = 1 »f,  50, and E» = 5 volts.
(a) For xEu, = 0.75 volt, plot to scale the waveshapes seen at both collectors, at
the base of T t , and at the common emitter.
312
SWITCHING
[Chap. 10
Fio. 1015
(6) Repeat part a for xEu, «■ 1.5 volts. Superimpose these plots on the wave
shapes drawn for part a.
102. We wish to adjust the multivibrator of Prob. 101 so that it will generate a
2volt 10msec pulse at the collector of T t . This should be the maximum possible
pulse width which can be produced by this circuit.
(a) Specify the new values of R», x, and C needed to satisfy these conditions.
(6) Sketch the waveshape appearing at the base of TV
103. (a) Calculate the limits on x,
+ 10v separating the various modes of opera
tion for the multivibrator of Fig. 1015.
Select a value of x that lies halfway
between the extremes of mode 12, and
calculate the pulse duration and ampli
tude at the collector of T% (assume Rt
small).
(6) Solve for the setting of x that will
produce a pulse of the same duration but
in mode 32. Plot the collector wave
shape to scale on the same graph as in
part a.
104. Design an emittercoupled multi
vibrator that will generate at its maximum
setting, a 10msec pulse. This circuit should switch directly from mode 12 to mode 32
and must operate in mode 12 over the widest possible range of x. Base your design
on a transistor having /8 = 50 and E» = 5 volts. Set Rt = 1 K. Specify the range
of x for both operating modes. Sketch the various circuit waveshapes produced when
the multivibrator generates its 10msec pulse.
105. (a) The multivibrator of Fig. 1015 is controlled by a 6volt battery in place
of the potentiometer. Ett suddenly drops by 5 per cent, from 10 to 9.5 volts. What
effect will this have on the pulse duration?
(6) If this same control voltage is derived by setting the potentiometer at
x = 0.6, by how much will the pulse width change when En is reduced by 5 per
cent?
(c) Repeat parts o and 6 if a 2,000ohm resistor is inserted in series with the base
of T t .
106. (a) Show three methods of triggering the multivibrator of Fig. 1015. Dis
cuss the required source impedance of the signal generator and the means of coupling
the pulse into the circuit.
(6) If a pulse is applied through a diode to .the commonemitter terminal, calculate
the required amplitude at x — 0.2, 0.4, 0.6.
107. (o) The emittercoupled multivibrator shown in Fig. 1015 must be adjusted,
by changing Ri, to generate the longest possible pulse. If x = 0.4, find the required
value of fli and the width of the pulse produced.
(6) With Ri = 1 K and with x set to give the maximum possible swing at the collec
tor of Ti, sketch and label fully the waveshape at the collector of Ti.
(c) Sketch and label the waveshapes at both collectors when x = 0.33 and when
/Si = 1 K.
108. (o) Plot the collector voltage swing of 7: as a function of the potentiometer
setting for the multivibrator described in the text in Example 101. Label all modes
of operation.
(6) Repeat the plot, on the same graph, for the collector voltage of Tu
14
Shaft
EMITTERCOUPLED MULTIVIBRATORS 313
109. The multivibrator whose characteristics are plotted in curve 1 of Fig. 107
is used in a special instrument to measure small linear displacements. This informa
tion is contained in the width of the pulse produced once the multivibrator is triggered
with a readout puke. For the transducer we employ a small capacity, replacing C
in Fig. 101, whose effective plate spacing is varied by the v ,. ,
shaft displacement (Fig. 1016). Assume that the minimum ^ I
value of C is 1,000 npi for d = 10 4 cm and that its maximum % '
value is 10,000 prf for d = 10~ s cm. Moreover, to prevent %
arcover, Eu is reduced to 2 volts. %
(a) When x =0.3, calculate the range of pulse widths %
generated as the rod is displaced. M
(6) An angular rotation $ of ±3° varies x between the „ „ ,
limits of 0.2 to 0.4 from its nominal value of 0.3. Under
these conditions, express the pulse duration < as a function of d and 0.
t, =f(d,B)
Evaluate all constants. Assume a linear variation of h with respect to both d and 8.
Is this assumption justified?
1010. (a) Verify the bounds of each mode for the multivibrator discussed in the
text in Example 102. Assume that R, remains constant at 2,500 ohms. Is this
approximation justified?
(6) Prove that the x boundary between mode 22 and mode 32 is independent of the
potentiometer employed for control.
1011. In the circuit of Fig. 1015, R d is a 5,000ohm control. Plot the waveshapes
at each transistor element, to scale, when x = 0.5. How does the pulse duration com
pare with the one generated when R d is very small?
1012. The equation
y = In ^ < x < 0.75
1 — x
defines the operating path of a circuit.
(a) For what value of x is y most nearly a linear function of x?
(b) If x is varied by ±0.05 about this point, what is the nonlinearity of y?
(c) What is the NL of y if x varies by ±0.05 about the point x = 14?
(Define the NL of y as the maximum variation in y from a straight line drawn between
its bounded end points divided by the change in y between the same two end points.)
1013. (a) Derive the limits of Eq. (1028).
(6) Find the numerical values of x that delineate the modes of operation of the
multivibrator discussed in Sec. 104. Tabulate, under each mode, the normal and
the switched state of each transistor. Are any modes of operation monostable?
Explain.
1014. Repeat the calculations for the sample problem given in Sec. 104 when x is
increased to 0.5. Make all reasonable approximations.
1016. Determine the amplitude and duration of the output taken at the plate of T*
in Fig. 1012 if x = H, Ri = fi 2 = 10 K, R, = 5 K, R = 500 K, C = 1,000 M,
r p = 20 K, fi. = 40, and En = 300 volts.
1016. (o) Find the minimum and maximum values of E t that still permit the
circuit of Fig. 1017 to function as a monostable multivibrator.
(b) With Ei adjusted to 5 volts below its maximum point, sketch and label the
waveshapes produced at et u ew, e t , and e„i when a trigger is applied at t = 0.
314
SWITCHING
[Chap. 10
?+300
10 K
Hf
10 K
(c) In which modes would the source impedance of Ei affect the pulse duration?
Explain your answer.
1017. The circuit of Fig. 1017 is modi
fied by returning the timing resistor to
ground instead of to En and by increasing
the capacity to 2 /if. Solve for the response
at all pertinent elements after a trigger is
injected, if Ei = 50 volts. Does this
I ^ i > h i modification improve or degrade the opera
[ + V3 Y ^— f tion? Explain.
^JBj I i 1 1018. The transistor multivibrator of
Ir=10K Sck ^ s " i"" 1 ^ ' s designed to operate at very
/«=20 ? low voltages. Consequently its small
_ re 500 i emitter base drop must be taken into
account in calculating the time duration.
Furthermore, the temperaturedependent
ho increases from 1 ^a at 20°C to 20 /ta at 100°C. Calculate the nominal pulse
duration and amplitude at the collector of T% (at 0°C). By how much will the pulse
vary over the expected temperature range ? For your calculations use the transistor
model of Fig. 10186.
0.01 nX
r p =10K
A=20
rc500
Fig. 1017
+1.5 v
bo
03+U/co
— o—
* 0'L
:=:o.2v
T
(b)
Fig. 1018
BIBLIOGRAPHY
Chance, B., M. H. Johnson, and R. H. Phillip: Precision Delay Multivibrator for
Range Measurements, MIT Rod. Labs. Rept. 632.
Clarke, K. K., and M. V. Joyce: "Transistor Circuit Analysis," AddisonWesley
Publishing Company, Reading, Mass., in press.
Glegg, K. : Cathodecoupled Multivibrator Operation, Proc. IRE, vol. 38, no. 6, pp.
655657, 1950.
Millman, J., and H. Taub: "Pulse and Digital Circuits," McGrawHill Book Com
pany, Inc., New York, 1956.
Reintjes, J. F., and G. T. Coate: "Principles of Radar," 3d ed., McGrawHill Book
Company, Inc., New York, 1952.
Schmitt, O. H.: A Thermionic Trigger, J. Sci. Instr., vol. 15, p. 24, 1938.
CHAPTER 11
NEGATIVERESISTANCE SWITCHING CIRCUITS
A twoterminal active network that exhibits a negativeresistance
drivingpoint impedance over a portion of its range may, together with
an energystorage element, be used as a switching circuit. Our initial
discussion concerns a postulated ideal device in various circuit configura
tions. After considering several available active elements that are used
for this type of switching, we shall modify the original argument to
account for the frequency dependence of the nonideal active elements.
A physical device differs markedly from an ideal one, and some of the
latter results may even contradict the earlier conclusions. If the reader
considers only the problem posed in each individual network, the argu
ments employed will be found to be consistent.
111. Basic Circuit Considerations. The regenerative circuits treated
in Chaps. 9 and 10 — bistable, monostable, and astable multivibrators —
depended for their proper selfswitching action on the charge stored in a
capacitor (the single energystorage element of the circuit) maintaining
the tube or transistor cut off. Only after a sufficient amount of the
stored energy had been dissipated did the active element reenter its
active region; then, during recovery, the original stored energy was
replenished through a positive source impedance. But in order to trans
fer the capacitor from a region of low potential energy (off) to one of high
energy (recovery), the switch itself must contain an energy source. Thus
a switching circuit might well be characterized as a device having two
stable or quasistable states, i.e., the dissipative regions, and a single
transition zone.
Any twoterminal device with these attributes will, when combined
with a capacitor or inductance, function as a multivibrator. Figure 111
presents the idealized voltampere characteristic of one class of such
devices. In regions I and III, the curve has a positive slope with a cor
responding positive incremental resistance. These are the dissipative
segments. Region II is the negativeresistance portion of the character
istic, and it is here that the switching of circuit states takes place.
Negativeresistance characteristics arise only through the use of active
elements, which serve as the source of the available energy. Passive
315
316
SWITCHING
[Chap. 11
Fig. 111. Idealizednegativeresistance
characteristic (currentcontrolled) .
elements, by their very nature, are always dissipative. Yet no physical
device can be expected to have an unlimited range of negative resistance;
to do so implies an infinite source of energy. Therefore each negative
slope will always be flanked by two regions of positive resistance.
Because the terminal voltage is a singlevalued function of current, we
classify the characteristic of Fig. 111 as a currentcontrolled nonlinear
resistance (CNLR). On the other
hand, any one of three values of cur
rent becomes possible upon exci
tation by an input voltage. The
particular value that will flow is
determined by the constraints im
posed by the external circuit, with
this very multivaluedness permitting
the several different circuit states.
Since only the current is unre
stricted, in our switching circuit we must utilize an energystorage ele
ment allowing instantaneous current changes during the switching process,
i.e., a capacitor.
An alternative form of the nonlinear resistance, the dual of the above,
also exists. Its voltagecontrolled characteristic is illustrated by the
sketch of Fig. 112 (VNLR). To complete the duality, an inductance
would be used instead of the controlling capacitor to permit rapid
terminal voltage changes.
Negativeresistance regions are exhibited by many devices. Current
controlled characteristics appear in pointcontact transistors, in uni
junction and avalanche diodes, and
in gas tubes. For examples of the
voltagecontrolled nonlinear resist
ance (VNLR), we may refer to the
screen characteristics of a tetrode or
a pentode, to the tunnel diode, and
to the pnpn transistor. In addi
tion, the use of feedback allows the
production of almost any desired
form of NLR. Generally voltage
feedback furnishes the voltagecon
trolled characteristics and current feedback establishes those that are
currentcontrolled.
Behavior of the two classes of NLR are similar enough so that any
conclusions derived from the examination of one class may be applied to
the other. Of course, caution as to the duality relationship must be
observed while interpreting the results.
Fio. 112. Voltagecontrolled nonlinear
resistance.
Sec. 112] negativeresistance switching circuits
317
112. Basic Switching Circuits. The basic form of the switching cir
cuit with which we shall be concerned is shown in Fig. 113. For the
negativeresistance element, we will use the currentcontrolled character
istic previously given. In this circuit, the items of interest are the neces
sary conditions for stable operation at any one point and the mechanism
of transition from one state to the other. We shall start the analysis by
writing the singlenode equation :
tr = i n + ic = in + C
dv n
dt
(111)
and V nn  i r R = v n (112)
Solving Eq. (112) for i, and substituting the result into Eq. (111) yields
,dv„
Let
(V nn  i n R)  », = RC
v t =RC^
at
dt
(113)
(114)
However, F„ n — i„R is simply the static (dc) load line which we can
superimpose on the voltampere characteristic. The distance measured
h 'n
« 4
t
CNLR
r *nn
~°
.
"»
B
A i
/c
K.A/
s"'
'x
^°
V„„\
/ '"
Eh
^^VnninR
^^ Load line
Fig. 113. Basic CNLR switching circuit. Fig. 114. Graphical solution of the cir
cuit of Fig. 113.
from the known curve to the load line must be v ( , and as shown in Fig.
114, it may be treated as a vector.
The steadystate operating point, the intersection of the load line
and the characteristic, is stable if, and only if, any momentary perturba
tion will create the conditions forcing a return to the original point.
Since the voltampere plot represents the locus of all possible behavior of
the NLR, any disturbance will shift the operating point along it, say
from A to A'. Equation (114) defines the vector v ( , drawn from any
point on the characteristics to the load line, as being proportional to the
time rate of change of v n . At point A', this vector is negative, indicating
that the terminal voltage across the capacitor must decrease with time.
If instead the initial disturbance were from A to A", the vector would be
positive, with the terminal voltage increasing with time toward point A.
318
SWITCHING
[Chap. 11
Only at A itself is v 4 zero, and only here do we have a stable operating
point.
Operation as a Monostable Switching Circuit. A voltage pulse, applied
in series with the bias supply, momentarily shifts the load line to the
position indicated by the dashed line of Fig. 115. While the pulse is
present, the shifted load line establishes a new stable point at F. But v h
as drawn from the original operating point A to the shifted load line, is
positive, and therefore the terminal
voltage increases toward point B.
At the peak, vt is still positive, for
bidding travel down line segment
BD. Operation must be restricted to
the characteristics and must stabilize
at point F. It can do so only if the
operating point can transfer from
segment EB to CD. The voltage
across the capacitor may not change
instantaneously, but the current can and does; it jumps from B to C.
Along line segment CF, v< becomes negative. After the jump, the
direction of the operating path is down toward point F.
Removal of the input trigger permits the load line to return to its
original position. Even after it does so, the terminal voltage, constrained
by the capacitor to change slowly, remains on segment CD and travels
toward point D. At D, v ( is still negative, preventing the traverse up the
line segment DB. A second jump in current is indicated, this time from
D to E. Following this jump, v ( becomes positive, and the operating
path is finally back toward the original stable point A.
In order to repeat the operating cycle, another trigger must be injected.
The minimumsize pulse, for guaranteed triggering, is that value neces
sary to shift the load line so that it just clears the peak point B.
As an alternative method of triggering, we can inject a current impulse,
having an area of K ampsec, into the common node of Fig. 113. KS(t)
changes the voltage across the capacitor from the initial value of Va to
Fig. 115. Monostable operation.
»(0+)
Va + —
If y(0+) is greater than the peak voltage V B , the operating point will
switch to the appropriate voltage coordinate along the extension of line
segment CD. Recovery proceeds as before, from the new initial value
established by the applied impulse.
Since this system has one, and only one, stable condition, any disturb
ance, no matter how large, eventually results in the circuit returning to
this point; it functions as a monostable device.
Sec. 112] negativeresistance switching circuits
319
Bistable Operation. For operation as a bistable, the value of V„„ and
the load resistance will have to be changed so that the circuit has the two
stable points (A, G) indicated in Fig. 116. Upon first examination, H
appears to be a third stable point, but once switching begins, the transi
tion will always be from A to G for positive triggers and from G to A when
a negative trigger is applied. The operating path is in the direction of the
arrows. Regardless of how or where the triggers are injected, it is
impossible to establish a transition path to point H.
A practical CNLR differs in important aspects from the ideal, and it is
these very differences that preclude the establishment of point H as a
third stable point (Sec. 118). In general, it is difficult to stabilize a
system within its negativeresistance region; whenever there is an appre
ciablesize energystorage element present, it becomes almost impossible
to do so.
B
"" /
y^^H
JJ. /
/ °
eA — •
D
Fig. 116. Bistable operation.
Fio. 117. Astable operation.
In effecting the change of state, the external trigger forces the operating
point to move from its stable position to the peak of the curve, i.e.,
from A to B or from G to D. In sweeping through this distance, power
proportional to the product of the changes in voltage and current will be
dissipated, and this power must be supplied from the trigger source.
We conclude that if the stable points are established closer to the peaks,
the switchingpower requirement imposed on the trigger generator will
be much less severe.
The sequence of the multivibrator's behavior in switching from one
stable point to the other is identical with that discussed for the mono
stable circuit.
Astable Operation. When the load line is so chosen that it intersects
the voltampere characteristic only on the negative portion of the NLR,
no stable point of operation exists. This case is illustrated in Fig. 117.
A power balance, evaluated in the vicinity of the point of intersection J,
would show that the energy supplied by the NLR is greater than that
dissipated by the load resistor. The surplus energy, available for storage
320 switching [Chap. 11
by the capacitor, results in a voltage buildup toward either point B or
point D.
Once the peak is reached, the path of operation becomes BCDE, with
the instantaneous current jumps taking place from B to C and D to E.
We might observe that at all points on the characteristic, the value of v,
is such as to direct the time rate of voltage change toward the two peaks.
An alternative way of viewing the buildup phenomena is to recognize
that in region II the negative resistance — R n and the load resistor R are
in parallel. Surplus energy means that the parallel combination is still a
negative resistance with the system's pole located at
v =  C0F1F*3 (115)
This pole lies on the real axis in the right half of the complex plane.
Obviously, the circuit is unstable, having an increasing exponential for
its transient response. Terminal voltage builds up until limiting occurs
when the operating point enters the positiveresistance region.
Instantaneous current jumps are forbidden in any physical CNLR
circuit by the inductive component of the drivingpoint impedance.
This modifies the path of operation, introducing the curvature shown
by the dashed lines of Fig. 117. For verylowfrequency operation, the
transition time is small enough so that the path approaches the ideal.
But as the generated pulse duration decreases, the transition time
becomes an appreciable portion of the total cycle and the locus constricts
to the dashed curve sketched. At very high frequencies, the path may
even turn into a small ellipse.
Stability of Ideal CNLR Switching Circuits. For absolute stability,
i.e., the system's pole remaining on the lefthand half of the real axis, the
parallel combination of the ideal negative resistance and the external load
resistor must be positive.
_ R(R n )
Kp ~ R + {~R n ) >U
In the particular circuit of Fig. 113, where the graphical solution is
given by Fig. 117, this condition can be satisfied only when
R < \Rn\ (116)
If the load is a short circuit, then regardless of the value of R n the parallel
combination will always be positive. In addition, the now horizontal
load line intersects the positiveresistance, as well as the negative
resistance, region of the CNLR and consequently establishes at least
two stable operating points. Alternatively, an infiniteresistance load
Sec. 113] negativeresistance switching circuits
321
could intersect only the negativeresistance portion, and it would cause
astable behavior in the presence of any external capacity. It thus seems
logical to refer to the ideal CNLR as a shortcircuit stable device. Other
considerations will cause us to modify this conclusion for a physical
CNLR, and reference should be made to Sec. 118, where the topic of
stability is treated in much greater detail.
113. Calculation of Waveshapes. The calculation of the waveshapes
of an ideal switching circuit (no parasitic energystorage elements pres
ent) may be greatly simplified by
representing each linear region of the
NLR by a resistor in series with a
battery. If we extend the straight
lines of NLR until they reach the
voltage axis, the point of intersection
is the battery voltage and the slope
of the line segment is the resistance
value. Figure 1 18 shows the result
ant equivalent circuit for region I.
Since the instantaneous current jump carries the operating point through
the negativeresistance region, only regions I and III need be considered.
Within each region, the exponential terminal response has, as its single
boundary condition, voltage continuity across the current jumps of the
circuit. The problem resolves itself into finding the boundary values and
the time constants holding within regions I and III.
At the instant that the operating point enters region I, the voltage
across the capacitor is that of point E in Fig. 115, 116, or 117. It
rises with a time constant r t toward the steadystate voltage of the
TheVenin equivalent circuit across its terminals V\.
1 I' \
T V '
j_
Fig. 118. Piecewiselinear circuit used
for the calculation of waveshapes.
ti = C(ft [ ft,)
Fi = BV t + R,V n
fti + ft
(117)
(118)
This voltage [Eq. (118)] is simply the value of the intersection of the load
line with the extension of the line segment defining region I. In the
astable circuit, Vi must always be larger than V B ; otherwise the output
will never reach the voltage peak B. However, if the circuit is adjusted
for bistable or monostable operation, and if it has a stable point along the
segment EB, then 7 X will be the voltage at the stable point. Once this
value is reached, sweep action ceases.
At the peak, v n = V B and the operating point flips from B to C.
Upon entering the new region (III), the cireuit of Fig. 118 changes; ft m
and V IU replace fti and Vj. The terminal voltage immediately starts
322
SWITCHING
[Chap. 11
charging toward the new TheVenin equivalent voltage with a time con
stant T3.
RVm + RmV n ,
V 3 =
Riu + R
t 3 = C(R  Bar)
(119)
Equation (119) defines the final value of any circuit having a stable
point lying on line segment CD. For astable operation, V 3 < Vd
After reaching the peak, the operating path jumps to point E, and the
cycle repeats.
»E
Id
in
1
t
1
1
7 A
t * \
/ ' \
/ \ \
/ ' \
/ ! \
/ i \
r < \
/ %t
r
t
i
i
"^
^^
«3
t
~ii>m
^^"
«=Tj 
Fig. 119. Waveshapes produced in a CNLR switching circuit (astable operation).
The current values at the switching points are simply those given by
NLR characteristic at points B, C, D, and E. In order to find the steady
state values, we must consider the circuit when the capacitor is fully
charged and the full current flows into the NLR terminals. In region I,
and in region III,
h
/a =
' nn
 v 1
Ri
+ R
r nn
 Fin
#in + R
(1110)
(1111)
These values also correspond to the point of intersection of the extended
voltampere characteristic and the load line.
From the characteristic and from Eqs. (117) to (1111), we have
all the voltage and current values necessary to sketch the terminal
waveshapes. Figure 119 shows the current and voltage waveshapes for
Sec. 11^3] negativebesistance switching cibcuits 323
an astable configuration, and Fig. 11106 those appearing in the mono
stable circuit of Example 111.
The duration of the two portions of the generated waveshape, evaluated
by substituting the appropriate voltage values into the general time
equation, is
h = Ti In J'~ J/ (1112)
and U = t 3 In Z 3 ~ Y* (1113)
V z — V B
In some NLR devices we may exercise a measure of control over the
shape of the voltampere characteristic. Adjusting the ratio of Ri to Rm
determines the ratio of pulse width to pulse spacing. Furthermore, if Ri
is made very much larger than R m and if Fi is very much larger than V B ,
then the circuit becomes an almost linear sweep. Its fast recovery time
is mainly dependent on Rm.
We see that the greater the degree of control over the characteristic,
the more versatile the system. Not all devices allow for wide latitude of
adjustment, but even in those that do not, the characteristics might be
modified by external diodes biased for conduction at the desired voltage
levels (Chap. 2).
Example 111. A CNLR having the voltampere characteristics given in Kg.
ll10a is used as a monostable pulse generator. Its external load consists of a l/ti
capacitor and a 2,000ohm resistor in series with a 20volt source, as shown in Fig. 113.
Only two sets of coordinates are unknown: (1) the location of the intersection of
the load line with the line segment defining region III, i.e., the single stable point A;
(2) the intersection of the load line with the extension of segment I, that is, Vi and h.
In order to find these points we first solve the graph for the three denning equations
Region I: »„i = 110 + (5 K)i„
Region III: t,„ m  12.5 + 250i„
Load line: v nL  20  (2 K)i„
By equating »»m to t>„£, we find that the stable point is located at
1% = 14.45 ma and V, = —8.9 volts
The other set of coordinates is
1 1 12.85 ma and V, = 45.7 volts
After the pulse is applied, the operating point jumps to point F, and i n and v„ start
charging toward /, and V,. From Eq. (1112), the time required to reach the peak is
h = 1,430 X 10«In ^ + ^ = 636 ^ec
where t,  (2 K  5 K)l ^f = 1,430 Msec.
324
SWITCHING
[Chap. 11
Upon jumping to point C, the circuit recovers toward the stable point A with
the time constant
t, = (2 K  250)1 »i = 222 yusec
Figure 11106 shows the current and voltage waveshapes generated in this circuit.
Note that the recovery portion of the wave is longer than the pulse duration and that
the voltage rise is almost a linear sweep.
(20,10) K >
&/tsec
Fig. 1110. Monostable operation, (a) CNLR characteristic and graphical construc
tion for Example 111; (6) current and voltage waveshapes produced after triggering.
We might further note that the ratio of the recovery time to the
pulse duration depends on the relative slopes of the two positiveresist
ance regions. For a fast pulse and a long recovery, the single stable
point should be established in the highresistance region. For a relatively
fast recovery and a long sweep, the stable point must be placed in the
lowresistance region, as it is in the above example.
Sec. 114] negativeresistance switching circuits
325
114. Voltagecontrolled NLR Switching Circuits. The basic switch
ing circuit associated with the VNLR is simply the dual of the one dis
cussed in Sec. 112. Figure 1111 illustrates the resultant series switch
ing circuit together with the graphical construction holding when adjusted
for monostable operation.
By properly arranging the terms, we can write the input loop equation
in a form that will present the steadystate solution as well as indicate
the permissible time variation.
(Vi.«)».=L^v.
(1114)
It follows from Eq. (1114) that the intersection of the load line (V nn —
i„R) with the characteristics «„ is the only possible stable operating point
(a)
Fig. 1111. (a) Basic VNLR switching circuit; (6) monostable path of operation.
(di„/dl = 0). Any perturbation from this point creates a time rate of
change of current that forces the operating point back to the original
stable point (point A in Fig. 11116). Thus the path of operation,
location of the stable points, calculation of minimum trigger amplitude,
and the calculation of circuit waveshapes are similar to those discussed
for the CNLR. The major difference is that now the current is con
strained to change slowly, but this only supports the duality relation
ship, i.e., the interchange of the circuit's voltage and current response.
The second important difference in the ideal VNLR concerns the
condition which must be satisfied for absolute system stability. Here
the external load appears in series with any internal incremental resist
ance. In order to prevent the system pole from migrating into the right
half plane, the total series combination must remain positive.
or
R. = R + («,) >
R > Rn
(1115)
This stability condition holds only for the ideal VNLR circuit. A
practical device differs in important respects from the ideal, and as with
the CNLR, the stability criterion will be reexamined in Sec. 118,
326
SWITCHING
[Chap. 11
115. NLR Characteristics — Collectortobasecoupled Monostable
Multivibrator. As an example of how all singleenergystorageelement
switching circuits may be treated from the viewpoint of their NLR
characteristics, we shall now reexamine the collectorbasecoupled mono
stable multivibrator of Chap. 9. Since the logical place to look for the
negative resistance is across the energystorageelement terminals, Fig.
ll12a shows the basic circuit with the timing capacitor omitted. In
our analysis we shall make use of the equivalent circuit drawn in Fig.
1112&. Both transistors are shown in their active region, and as each
is driven into either cutoff or saturation, the model must be accordingly
modified.
+25 v
(a)
lO* 3 '"
R
•vVSA
WV
\ZVV I I
(b)
Flo. 1112. (a) Monostable multivibrator; (6) equivalent circuit. The timing
capacitor is normally connected across the drivingpoint terminals.
The circuit is initially adjusted so that in its normal state T 2 is fully
on and T\ is cut off. Even though the storage time of the saturated
transistor limits the switching time, this configuration is considered
because of its inherent simplicity. Further simplification arises from the
following conditions:
1. /S is very large; therefore /S + 1 = 0.
2. R » R 2 , and R 3 » R 2 .
Generally, three or four line segments are sufficient to define the com
plete voltageampere characteristic. In finding these segments attention
should be focused on the controlling current, i a . By starting from a high
positive value of input current and considering the permissible change of
state as the current decreases to, and below, zero, the possible operating
regions are:
Sec. 115] negativeresistance switching circuits 327
Region 1 Tt saturated and 2\ off
Region 2 Both transistors in their active region
Region 3 Tt active and 2\ saturated
Region 4 Tt off and T\ active
• Region 5 T s off and Ty saturated
At most, two of the last three regions can exist. Once T\ saturates,
the decreasing input current may still drive Ti through its active to its
cutoff region. Thus there is the possibility of operating over regions 3
and 5. On the other hand, if T 2 is driven into cutoff while Ti is still
active, the final bounding region will be number 4.
In each region, we shall solve for the equation of e„ as a function of i„,
in the form
e„ = E { + R,4a (1116)
Ei is the voltage intercept found at i a = 0, just as if the defined region
remained invariant. Ri is the incremental drivingpoint resistance in the
region under examination. A sequence of such straight lines presents,
in a piecewise manner, the complete voltampere characteristic. In order
to find the boundaries of the individual segments, we solve for the coordi
nates of the intersection of the appropriate two lines. We might also con
sider the physical limitations imposed by the active elements employed
in establishing these bounds.
Region 1 — T 2 Saturated and Ti Off. Here 4i = and e C 2 = 0, and
hence
e a = Eu, + i„R! (1117)
The lower limit of this region, where both Ti and T 2 become active,
occurs once
e C 2 = E»  PhtRi = (1118)
where «« = ^ + i a (1119)
From Eqs. (1118) and (1119) we find that this zone is limited to
• ~» ^ bb _ El*
%a > $Ri R
And since R < /3i?2, region 1 holds down to some negative value of i a .
Region 2 — 2*i and T 2 Active. From the solution of the circuit's loop
equation, when in is given by Eq. (1118) and ibi = e e %/R%,
Note that once a transmission path involving the gain of both transistors
exists, the slope of the voltampere characteristic becomes negative.
328 switching [Chap. 11
Here the positive feedback provides the required energy source. This
negativeresistance region is bounded when Tj turns off at iu = 0. And
the corresponding constraints on t« are
e» _ E& > «■ > _ Ea
jiRz R R
(1121)
Region 3 — Ti Active and Ti Saturated. The conducting collectorbase
diode of T\ and the emitterbase diode of Ti now shortcircuit the input
terminals. The input equation becomes
/
(1122)
Region 4 — Ti Off and Ti Active. Under the conditions holding in this
region,
ibi =
R%
Ri
and therefore e a = fiE» ^ + i a (R + Ri)
Tx finally becomes saturated for
(1123)
\Rt R\J
Region 5 — Ti Off and Ti Saturated. Ti being an open circuit and Ti
a short circuit, the defining equation becomes
e a = Eu, + i*R
(1124)
Example 112. In this problem we shall calculate and plot the characteristic for
the typical transistor monostable multivibrator of Fig. ll12a.
As the first step, the defining equation for each region is evaluated :
Region 1.
Region 2.
Region 3.
Region 4.
Region 5.
e. = 25 + (1 K)t«
e„ ^ 36.6  (81 K)i.
e„ =
e. = 29.2 + (51 K)i„
e„ = 25 + (50 K)i a
Eq. (1117)
Eq. (1120)
Eq. (1122)
Eq. (1123)
Eq. (1124)
By plotting the five straight lines, as in Fig. 1113, we can determine the permissible
circuit states. As the current decreases, we note that the segments traversed are
1, 2, 3, and 5. There is no possible way to reach line segment 4, and it therefore repre
sents a forbidden mode of operation for this particular circuit.
One of the two remaining coordinates of interest is the location of the valley point.
To find the intersection of lines 1 and 2 we equate (1117) to (1122), with the result
T 36.5  25
' " (81 + 1) K
V,  25.14 volts
0.14 ma
Sec. 116] negativeresistance switching circuits 320
The current intercept of region 5 is found from Eq. (1124) by setting ft, ~ 0,
7 " = oi! =  a5ma
All the resistors have been included in the base amplifier, and consequently the
voltage axis is also the steadystate load line. The circuit remains at its single stable
point (i« ■= 0, «. = 25 volts) until a trigger forces it into its unstable state. Transition
is directly from region 1 to region 4. Here the slope is simply the timing resistance
t
.Qp\
wits
i i
i i
/ i
i
i
i
//
i
i
/
/
i y
•3
*a.t
na
1
00
8o'y\
0.4
0.2
2
4
6
8 1
10
—15
• 20
4,
/
7*
\2
/ I
w
/
h
ivK
25
1
T
\
\
\
30
~ Sl(
)pe 1
0001
1
\
\
\
FlO. 1113. CNLR characteristics of the collectorbasecoupled monostable multi
vibrator of Example 112.
(50 K), and it is here that the output pulse is generated. When the circuit switches
back to segment 1, it enters the lowresistance (1K) recovery portion of the cycle.
Since the load line is the voltage axis, the time constants are the product of the
slope of the line segment and the timing capacitor. Moreover, the steadystate
charging voltage is the voltage intercept; the steadystate current must be zero.
All the information necessary to calculate the voltage across, and current through, C
is contained in the voltampere characteristic of Fig. 1113.
116. Some Devices Possessing Currentcontrolled NLR Character
istics. The Pointcontact Transistor. Illustrating, as it does, the com
plete range of problems which must be treated in applying a practical
330
SWITCHING
[Chap. 11
negativeresistance device to switching circuits, the pointcontact tran
sistor is the first device exhibiting CNLR characteristics with which we
are concerned. This was not only the first semiconductor device in
which a negativeresistance region was observed but also the first device
which was extensively used in twoterminal switching circuits.
Positive current feedback from the collector back to the base, further
emphasized by padding the base with a large series resistor, produces
a negative drivingpoint impedance at either the emitter or collector
terminals, with the most useful characteristic appearing at the emitter.
The transistor circuit and its largesignal equivalent model are shown
in Fig. 1114.
First the lowfrequency, static drivingpoint characteristic may be
evaluated by examining the three possible regions of transistor operation,
<ti t
Ecc^:
Fig. 1114. (a) Pointcontact transistor connected to develop a CNLR drivingpoint
impedance; (b) largesignal T model.
cutoff, active, and saturated. Under cutoff conditions (i e < 0) the cir
cuit model would be modified by removing the current generator, by
replacing the emitter diode with its reverse resistance r re , and by replacing
the collector diode by its large reverse resistance r„. From the reduced
equivalent circuit, we find that the linear equation defining this region is
v, = ie[r r . + (r b + flx)  (r« + jR 2 )] 
r b + Ri
n + r„ + Ri + Rt
S i.r r ,
2?cc
(1125)
where r„ :» (r 6 + R x )  (r« + Ri) and where the polarity of E cc is taken
into account (Fig. 1114). The large value of r rc appearing in the
denominator usually makes the value of V p close to, but slightly less than,
zero.
In the active region, the emitter diode is forwardbiased, and we might
conveniently replace it by a short circuit. Also, r rc >J> (Ri + #2 + rt),
allowing the representation of D c by an open circuit for the purpose of
computing the input impedance. As a consequence, the drivingpoint
Sec. 116] negativekesistance switching circuits 331
equation becomes
v, £* i.{\  a){n + Ri)  V p (1126)
where V p is the voltage intercept at t, = 0. Within this region the
regeneration that develops the negative drivingpoint impedance is
supplied by the greaterthanunity value of the emittertocollector
current generator (a > 1).
The final zone, complete transistor saturation, is characterized by
heavy conduction through both diodes shown in Fig. 11146. For the
defining equation we need only consider Ri, r b , R it and Ea,.
v. = [(r 4 + flx)  RJi.  ■ ri +z 1 + Rl E « C 11 " 27 )
To find the coordinates of the valley point, i.e., the point separating
the active from the saturation region, we simply solve Eqs. (1126)
and (1127) simultaneously. But before beginning the calculation, even
further simplification is possible. We are primarily interested in the
negativeresistance portion of the characteristic, and in order to ensure
an appreciable negativeresistance value, Ri !2> n [Eq. (1126)]. If
we also neglect the small value of V p in Eq. (1126), the location of the
valley point will be given by
T ~ ~ E " V ~ ~ Rl ^ ~ °i) Ecc m^S"!
U = fli(l  a)  aR 2 K "J?i(l  a)  «R 2 K '
One obvious design difficulty is that the valleypoint coordinates are a
function of the transistor a and vary rather widely between individual
transistors. We can compensate by making Ri adjustable.
Figure ll15a shows the three regions of circuit behavior together
with their particular resistance values. In a practical transistor, the
nonlinear characteristics would introduce some curvature at both peaks.
The readily accessible additional elements of the device used to gen
erate the CNLR characteristic permit the extraction of an output
waveshape at a point well removed from the pulse timing network of
R and C (Fig. 113). If the output is taken at the collector, with the
circuit so adjusted that the critical timing occurs during the cutoff
interval, then any loading of the collector will have only an extremely
small secondorder effect on the pulse duration.
Within the active region the proportionality between collector and
emitter current is simply a. Outside this region, the transistor is either
cut off, with the collector current dropping to its small 7 c0 value, or
saturated, with the incremental current relationship now becoming
ic^s^pi. (1129)
til T til
332 switching [Chap. 11
To this we would add the current contributions from the bias batteries,
which of course depend on the particular circuit configuration. But since
we already know the location of the valley point of i, [Eq. (1128)],
the corresponding collector current is simply
I„ = al v
For current flow above this value, the slope of the transfer characteristic
is given by Eq. (1 129). The information we now have yields the current
transfer relationship shown in Fig. 11156.
«c
,
iu ie
y»
*«r
(b)
3%
Fig. 1115. (a) Pointcontacttransistor CNLR drivingpoint characteristic and (6)
emittercollector current transfer characteristic.
Since the transistor switches through the active region very rapidly,
when we evaluate the collector output waveshapes (both current and
voltage), only the cutoff and saturation regions need be considered.
This calculation will be left to the reader in his solution of individual
problems. Furthermore, the additional elements permit the injection
of the input trigger at the transistor base. By taking advantage of the
amplification afforded within the switching transistor, the trigger require
Sec. 116] negativeresistance switching circuits
333
merits become much less severe. And under cutoff conditions the trigger
source is also isolated from the timing circuit.
The pnpn Junction Transistor. The pnpn junction transistor also
exhibits CNLR characteristics (Fig. 1116), and like the pointcontact
transistor, it depends for its negativeresistance region on the positive
current feedback from the output back to the base. In fact, within the
"i
Fig. 1116. pnpn circuit and its CNLR characteristic.
pointcontact transistor a pn junction is established between the collector
contact and the base, thus leading to a pnpn structure. This similarity
allows the immediate application of all results derived for the point
contact transistor to the new device.
We might alternatively represent the pnpn transistor by a coupled
configuration of two junction transistors, a pnp and an npn (Fig.
1117). When both are within their active regions, the collector cur
rent of T\ is injected into the base of T%, where it is amplified by the
11 «JU * J
J 1$
"1
(!«)», pah \ l fl i
Fig. 1117. The twotransistor equivalent configuration of the pnpn circuit. Junc
tion currents are indicated.
large basecollector amplification factor 0. Finally, the collector of T t ,
directly coupled to the base circuit of T\, supplies the positive current
feedback necessary to develop the negative input impedance. The
approximate current flow in each branch of Fig. 1117 is indicated in the
interest of clarity. Note that the total composite base current is nega
tive. By substituting the value of 0, we find this current to be
in = in — id = [! — (« + P<*)]ii = (1  0)ti
(1130)
334
SWITCHING
(Chap. 11
Except that the larger amplification factor replaces a, the current
given by Eq. (1130) is of the identical form as the current that would flow
in the base circuit of the pointcontact transistor (Fig. 11146). Thus
the input impedance of this circuit, when it is in its active region, may
be expressed as
B ta S (1  fi)Ri (1131)
The cutoff frequency of this transistor can be made quite high, allowing
satisfactory operation in switching circuits generating ljusec or smaller
pulses.
The Unijunction Transistor. A silicon unijunction transistor consists
of a bar of n material with a p emitter junction located somewhat above
the center point of the bar (at the 60 to 75 per cent mark). Under the
emitter cutoff condition, the conductive current flow through the high
resistivity silicon establishes a voltage drop from the emitter to base 1
Fig. 1118. Unijunction transistor circuit and input voltampere characteristic.
proportional to the spacing between these points. The emitter remains
backbiased until the input driving voltage exceeds the conductive drop.
Once the emitter begins conduction, the minority carriers injected into
the n bar travel to base 1 and increase the charge density in this region;
they effectively decrease the drivingvoltage drop required for a given
current flow. Consequently, a negative input resistance appears between
the emitter and base 1 (Fig. 1118).
An equivalent circuit for the unijunction transistor which explains, at
least to a first approximation, the behavior of this device in its various
operating regions is presented in Fig. 1119. While it remains cut off,
the effective input impedance is simply the reverse emitter diode resist
ance, about 500 K. But once the emitter voltage reaches the backbias
voltage yEu,, where
r»i
and
V, = q£»
the diode conducts. The defining equation now becomes
v. = nEtb + [r. + (1  y)r hl  r b2 ]i.
(1132)
Sec. 116] negativebesistance switching circuits
335
If, in addition, account were taken of the small voltage drop (0.7 volt)
appearing across the emitter junction, then the peak point of Fig. 1118
would be shifted slightly up and to the right.
The current amplification factor y varies with frequency in much
the same manner as does a in the pointcontact transistor. But y further
depends on the mobility of the
majority and minority carriers, and
therefore it will change with the
current flow, from a value of ap
proximately 3 at low current density
to unity at saturation. As a result,
the negativeresistance region is ex
tremely nonlinear. In the model
drawn in Fig. 1119, all regions are
linearized by assuming a constant y.
The firing of the diode D 2 charac
terizes the saturation region, and the
full generator current yi, flowing through it serves to maintain full
conduction. Replacing D 2 by a short circuit permits the writing of the
appropriate saturationregion equation:
v.= V.+ (R,  r 61  r„)i. (1133)
R. II r*i
Fig. 1119. Model describing the be
havior of the unijunction transistor.
where
V,
R. II r hl + n, 2
Eu
A large measure of control over the slopes and intercepts of the indi
vidual regions is afforded by the insertion of additional resistance in series
with bi and 6 2 . Any resistance we add to 6i has the most pronounced
effect on the saturation region, and that included at 62 mainly increases
the magnitude of the negative input resistance.
Currently available unijunction transistors are limited to lowspeed
switching applications by their y cutoff frequency of only 0.7 or 0.8 mega
cycle. The correspondingly large input inductive component of the
input impedance (Sec. 118) will adversely affect the switching path even
at low frequencies. Rather than the expected large instantaneous cur
rent jumps from the peak and valley, the path of operation becomes
almost elliptic.
Avalancheregion Operation. Among the other devices exhibiting
currentcontrolled negativeresistance regions are the spacecharge
diode and the avalanchebreakdown transistor. Their current multipli
cation depends on the high reverse voltage establishing an extremely
high electric field intensity. This field dislodges valence electrons, and
by creating additional electronhole pairs, it increases the current density.
Avalancheregion operation is widely used for very fast switching circuits
336
SWITCHING
[Chap. 11
since there is no inherent minority storage time. However, the negative
resistance region is quite narrow and the circuit will generate only a small
amplitude signal.
117. Some Devices Possessing Voltagecontrolled Nonlinear Charac
teristics. The tetrode, one of the earliest devices found to exhibit volt
agecontrolled negative resistance, was the predecessor of many of the
secondaryemission multiplier tubes in current use. Because the energy
supplied to the external energystorage elements often resulted in an
undesirable oscillation, operation within this region of the plate volt
ampere characteristic was usually avoided. Realizing now its usefulness
for switching and oscillator applications, and in order to round out our
discussion of NLR devices, we shall discuss the tetrode along with several
other voltagecontrolled devices. Electricalengineering literature refers
Fig. 1120. Tetrode VNLR plate characteristic.
to the particular tetrodes used for their negativeresistance character
istics as dynatrons.
At very low values of plate voltage, the total cathode current flows
to the screen, which is the tube element at the highest potential. As its
voltage is raised, the plate receives an everincreasing percentage of the
total current flow. The electrons attracted to the plate strike it with
more force, because of additional energy acquired as they are accelerated
through the higher potential present. Eventually, their energy on
impact becomes sufficient to cause emission of the valence electrons of
the plate material. Since the screen is maintained at a constant poten
tial, somewhat higher than that now appearing from plate to cathode, the
secondaryemission electrons will travel from the plate back to the
screen, causing the current at the plate to decrease, even though its
voltage continues to increase (Fig. 1120). Also present is the possibility
that the plate current may become negative as a consequence of the large
emission from the plate surface.
As the plate voltage continues rising, up to and even beyond the
screen voltage, the secondary electrons begin returning to the plate.
Sec. 117] negativeresistance switching ciecuits
337
Under these circumstances, the screen current falls as the rising current
at the plate bounds the negativeresistance zone with the expected
dissipative region.
Pentode VNLR Characteristic — The Transitron. A pentode may be
so connected that it also will display a VNLR drivingpoint character
istic, but at its screen rather than at its plate (Fig. 1121). At low
values of screen voltage, the suppressor, highly negative with respect
to the cathode, completely cuts off the plate current. The tube func
tions as a triode, and the total cathode current, which now flows to the
screen, obeys Child's %power law. In this region, the screen resistance
is identically that of the tube connected as a triode, r c2t . As the screen
voltage continues rising, the suppressor voltage E c2  E 3 which must rise
with it eventually reaches a point where it allows plate conduction. Any
Fio. 1121. Pentode circuit for VNLR transitron operation and resultant screencircuit
voltampere characteristic.
further rise in the screen voltage increases the total tube current, and
simultaneously, through its effect on the suppressor, E e2 increases the
percentage of the current flowing to the plate. Less current flows in
the screen circuit, and the voltampere characteristic now exhibits a
negativeresistance region.
Once the suppressor becomes positive and draws current, it saturates.
The current flow through R 3 maintains the suppressor voltage at only a
few volts positive with respect to the cathode. Above this point, the
ratio of plate to screen current remains reasonably constant, and both
now increase with the rising screen voltage. Since the conditions are
identical with those treated for the phantastron in Sec. 69, the screen
resistance in this region will be
r C ip = r cJl (l + p )
where r eit = resistance in first region considered
P = ratio of plate to screen current
If we were now to turn back to the phantastron, which was discussed in
Sees. 68 and 69, we would see that the switching action occurring in
338
SWITCHING
[Chap. 11
the screen circuit depends on its negativeresistance characteristic. Since
the screen configuration is essentially of the form shown in Fig. 1121,
the normal load line should be selected for bistable operation. The two
circuit states are (1) when the suppressor cuts off the plate (region I
of Fig. 1121) and (2) when the suppressor saturates (region III of
Fig. 1121). The normal position of the phantastron is in region I,
requiring a positive pulse for switching. However, the circuit itself sup
plies the negative pulse necessary to switch the screen back from region III
to region I. The presence of the varying controlgrid signal means that
the path of screen operation will be along the composite of several of the
family of characteristic curves, instead of along a single curve.
1
2.0
.
V
3
= .1.0
«7
M>
c
7
CO/
Y
5
1(
)0 1!
)0 2(
30 2
JO 3(
X) '
m
(a)
Fig. 1122. (a) Tunneldiode voltampere characteristic; (b) model holding in the
negativeresistance region — L, is due to lead inductance, R, represents the lead resist
ance and ohmic losses in the semiconductor, and C is the junction capacity.
Tunnel Diode. The tunnel diode, which was discovered by Dr. L.
Esaki in 1957, consists of an extremely thin pn junction formed between
two heavily doped regions (large amounts of added impurities) of a semi
conductor. The shape of the voltampere characteristic (Fig. ll22a),
which exhibits a voltagecontrolled negativeresistance region for small
forward biases, can be explained from a consideration of the electron
wave propagation through the junction boundary. Because of the
heavy doping there exist relatively large numbers of conduction electrons
in the n material and a wide range of empty states in the p material. The
electron wave propagates freely within each region but cannot tunnel
through the potential barrier at the junction unless the energy level on one
side is matched by an equivalent empty state on the other side. During
the transition the electron wave is attenuated, while the energy is con
served. Thus the junction must be thin in order for the electron wave
to have an appreciable probability of transmission.
Sec. 117] negativeeesistance switching circuits 339
When the diode is backbiased, the energy level of the electrons in the
n material is lowered below that of the free electrons in the p material.
The reverse current, which is completely due to tunneling, can increase
without limit. In this region, the voltampere characteristic looks
exactly like that of a conducting diode.
An applied positive bias increases the potential energy of the electrons
in the n material. As a consequence the forward current will continue to
increase until the complete range of free states in the p material is
matched by the tunneling electron waves. Any further increase in the
forward bias raises the energy level of the free electrons above that of the
empty states and is, therefore, accompanied by a decrease in the terminal
current. This is the negativeresistance zone. Eventually the increas
ing bias causes the injection of the minority carriers, the diode conducts
in the normal manner, and the current again increases with increasing
voltage.
The major advantage of the tunnel diode over all other negative
resistance devices is the high speed of the current transmission. The
velocity of propagation approaches that of light. There is, however, a
large capacity (20 to 60 n/ii) associated with the junction as well as series
resistance and lead inductance shown in Fig. 1122&. These parasitic
elements tend to slow the switching time. In spite of this, the large
value of negative conductance permits the generation of pulses having
rise times of less than 10 10 sec. Sinusoidal oscillation at frequencies in
excess of 4,000 megacycles is also possible (see Chap. 15).
It should be noted that the negativeresistance region is quite narrow;
the peak is normally located between 50 and 100 mv and the valley point
at 150 to 500 mv. Thus, if the junction is designed for large peak cur
rents, the negativeresistance value will be extremely low. For example,
approximately —2 ohms is measured in the active region of a gallium
arsenide tunnel diode having a peak current of 100 ma. When we exam
ine a diode whose maximum current is less than 1 ma, we observe that the
average negative resistance has increased to a few hundred ohms.
Since the impedance of the power supply is of the same order of
magnitude as the negative resistance, it is extremely difficult to bias
properly and to stabilize the verylowresistance devices in their negative
resistance region. Even a small amount of lead inductance may lead to
undesired switching or spurious oscillations.
A twoterminal tunnel diode does not afford the same flexibility in
obtaining an isolated output as do the threeterminal active devices.
Consequently care must be taken that the external load will not disturb
the operation of the timing or switching circuits. In order to minimize
possible interaction when multiple stages involving tunnel diodes are to
be interconnected, some type of unilateral decoupling must be used. We
340 switching [Chap. 11
may even be forced to associate a junction transistor with each tunnel
diode circuit. Because the actual rise is limited by the slowest stage,
many of the advantages of the tunnel diode are negated. In some cir
cumstances it becomes practical to employ fastresponding diodes to
decouple the individual switching circuits. Diodes whose conduction
is due to tunneling, but which do not exhibit negative resistance, are ideal
for this purpose.
Pointcontact and jMipn Junction Transistor. A VNLR driving
point impedance will also be developed at the base of the pointcontact
transistor and at the base of its junc
° c tion equivalent, the pnpn transis
tor. Instead of merely discussing
W
,t
4^
O  — *~~ this device qualitatively, we shall
JJ L define the complete circuit behavior
by following our usual procedure of
examining the operation with respect
Ecc~=~ to the increasing independent vari
able, in this case Vi. When the drive
voltage is highly negative, the tran
Fig. 1123. Model for VNLK driving s i s t r will be completely saturated.
£or. imPedanCe ° f point " contact tran " Since this condition is characterized
by heavy conduction through both
of the diodes in the model of Fig. 1123, the input node equation, which
is written by superposition, becomes
H = EJBi + EJ3* + v l (G l + G 2 ) (1134)
where Gi = l/i?i and G 2 = 1/Ri.
As »i rises and as the transistor enters its active region, the emitter
current decreases with increasing driving voltage.
i. = (E M  »i)Gi
The input current contribution from E cc that normally flows through
r rc will be small enough so that it can be ignored, thus allowing the
following defining equation to be written for the active region:
ii (1  a)i. = (1  a)Gi(E m  »i) (1135)
From Eq. (1135) we conclude that a > 1 is the only required criterion
for a negative drivingpoint impedance at low frequencies.
When the pnpn transistor is used in place of the pointcontact
transistor, (3 would replace a in Eq. (1135).
Once the high positive value of Vi cuts off the transistor (vi > E„),
the circuit enters into its third region. With r„ > R\ and r« > R%, the
Sec. 118] negativeresistance switching circuits 341
input equation is of the same form as Eq. (1134); only the conductance
terms must be changed to conform to the new conditions.
*i = E ee g„ + E cc g rl . + vxig,. + g rc ) (1136)
A plot of the three characteristic equations appears in Fig. 1124.
Fio. 1124. Drivingpoint voltampere characteristic of pointcontact transistor— base
input.
118. Frequency Dependence of the Devices Exhibiting NLR Charac
teristics. Up to this point we have discussed only the static behavior of
the various devices that exhibit negativeresistance regions. But to
utilize these elements properly, their frequency limitations must also be
known. Some of these were treated briefly in earlier chapters, e.g., the
rise and fall times and the minority storage time of the solidstate devices.
The major item which we must yet consider is the effect of frequency on
the transistor parameters.
In each case, the negative input resistance developed in the active
region depended on the current amplification of the active element
employed. Equations (1126), (1131), and (1132) defining the nega
tiveresistance regions are all basically alike. Of course, the controlled
currentsource parameter is a in one case and or y in the other two, but
this is simply a detail dependent on the particular device considered. In
all three, the term of interest is of the form
Rin = (1  a)Bi
where a is now used as a general current amplification factor.
The parameter most affected by frequency is this very amplification
factor. Equation (1137) expresses its approximate variation, and we
note that a decreases at a rate of 6 db per octave as w rises above the
3db point u c .
a(a>) =
«o
(1137)
1 + jw/o>c
The a cutoff frequency /„ usually ranges from 100 kc to 2 megacycles
in the unijunction and the pnpn transistors, up to about 100 mega
342
SWITCHING
[Chap. 11
cycles in pointcontact and junction transistors that aie specifically
designed for highfrequency operation.
If we substitute Eq. (1137) into the general negativeimpedance term,
the resultant expression becomes
5 in S (l 1 + 7o,/ w J
Bi
/ u c 2 a \ , .
= Ri I 1 ,; — ; ) + .?«
\ c R\cto
2 + to 2
(1138)
The first part of Eq. (1138) represents the resistive portion of the input
impedance, which will remain negative up to
&h = W e V a O — 1
Above this frequency, the various transistors can no longer sustain their
negativeresistance characteristics and are unable to supply any energy
to the external load. It follows that switching operations would have
to be restricted to the range of frequencies below <a h .
The second term contributes an inductive component to the driving
point impedance. First of all, this prevents any instantaneous current
jumps. Secondly, it may also resonate with the external capacity,
generating an almost sinusoidal
waveshape, provided that the cir
cuit is biased within the active
region.
Figure 1125 shows an equivalent
drivingpoint network which repre
sents the incremental behavior of
the device within its active region
and which, of course, also satisfies
Eq. (1138). At low frequencies,
the input inductance is effectively
Fig. 1125. Equivalent drivingpoint net a short circuit, leaving only the
work of a CNLR in the active region and nega ti ve resistance term. At ele
the external load. yated frequencies the pre sence of
the additional energystorage term complicates the network and forces us
into a reevaluation of the conditions to be satisfied for absolute circuit
stability.
Stability of the Nonideal CNLR Circuits. The problem of stability can
be neither considered nor defined without treating the complete circuit
before us. Instability simply means that sufficient surplus energy exists
in the system to produce an increasing exponential response in any
energystorage element present. But if the external network also con
tains dissipative elements, then their effect on the system may even
Sec. 118] negativeresistance switching circuits 343
change one having an energy surplus into one that is completely dissi
pative. The particular conditions necessary to ensure this happening
are those that produce absolute stability and prevent any regenerative
switching.
Consider, for example, the network (Fig. 1125) that represents the
input characteristic of the currentcontrolled devices of Sec. 116. The
external load consists of a parallel combination of R and C together
with a bias battery whose function is to set the intersection of the load
line and the NLR characteristic within the proper region. In the follow
ing discussion we shall assume intersection of the negative resistance.
Since the NLR is currentcontrolled, for stability the poles of the current
function must not be allowed to he in the right half plane. And this
corresponds to restricting the zeros of the impedance function [Eq
(1139)] to the left half plane.
RtRCLp' + [(«! + R)L + CRRfaoil  a„)]p
= + «oRi[R + g x (l  «„)]
(pL + aoRJipRC + 1) t 1139 }
In order to guarantee this restriction, all the coefficients of the poly
nomial of the numerator of Eq. (1139) should have the same sign. This
leads to the two inequalities Eqs. (1140) and (1141), which we must
satisfy to ensure absolute circuit stability.
R > Ri(l  oo) (1140)
Ri + R
RRi(ao — l)w,
C < Efl^V (H41)
The value of L given in the model of Fig. 1125 was substituted while
solving for the condition expressed in Eq. (1141).
^ When both inequalities hold, the system cannot operate as a switching
circuit. Once C is larger than the minimum value given by Eq. (1141),
and if Eq. (1140) is still satisfied, the circuit becomes freerunning.
At the bounding value of C, the zeros of the impedance function he
on the imaginary axis; for C less than this value, the zeros are in the left
half plane and the system is stable. As C increases, the roots move
into the right half plane toward the real axis (Fig. 1126). If the zeros
are complex conjugates lying close to the imaginary axis, the time
response of the astable system is sinusoidal. This particular mode of
operation will be discussed in Chaps. 14 and 15.
For bistable operation R must be less than the magnitude of the nega
tive resistance; i.e., Eq. (1140) must remain unsatisfied. The load line
344
SWITCHING
[Chap. 11
•will now intersect the positiveresistance regions of the voltampere
characteristics. Both zeros of the impedance function will always lie
on the real axis, one in the right half plane and the other along the nega
tive real axis. As C increases, both zeros move to the right, the negative
root approaching zero as a limit and the positive root approaching
Ph = Wc(«0 — 1)
for large C. Thus the switching speed, through the negativeresistance
region, is limited primarily by the frequency response of the active
element employed.
,.
rl
).7i
■I m axis
II
 1
P
1
^
1 1 1
^1,500/^f
1.5
>
,0.4
• (
).3
).2
).l
10,000
///jf
4,500
' 10,000
0.3 0.2 0.1
"0.1
*(*2
0.3)
0.4 1 0T5
06'
07 n
xin e
R e axis
1
1
0.2
l
1
0.3
l
fl!=lK
«=2
■u c = 10 6
i?=2K
1
0.4
I
1
0.5
rH
If
1
M).7
Fig. 1126. Path of the zeros of Z(p) [Eq. (1139)] as a function of C. Specific circuit
values are given above for astable operation.
The stability condition presented in Eq. (1140) is exactly opposite
to the condition found when we discussed the ideal NLR in the earlier
portion of this chapter [Eq. (116)]. This does not really contradict
the earlier discussion, since a short circuit or a low resistance across
the input terminals of the equivalent network of Fig. 1125 results,
essentially, in the inductively controlled behavior of the VNLR (Fig.
1111). Furthermore, when &>„»<» or when L = 0, that is, when the
circuit approaches the ideal, the stability conditions found from Eq.
(1139) will reduce to the single equation given in Eq. (116).
Sec. 118] negativeresistance switching circuits
345
We conclude that a physical negativeresistance device cannot support
a stable point within the negativeresistance region unless the two
inequalities of Eqs. (1 140) and (1141) are satisfied. Point H of Fig. 1 16
(the loadline intersection with the negative resistance) will, in general, be
unstable. Away from the vicinity of this point, the system rapidly
stabilizes at one of its two inter
sections in the completely dissipative
regions.
Stability conditions are seen to
depend on the particular circuit con
figuration. In order to avoid ambi
guity, it is much better to refer
always to the device used in terms
of its singlevaluedness, i.e., current
or voltage, rather than its conditions
for stability. Furthermore, when
any changes are made in the external circuit, the reader must resolve
for the conditions necessary to ensure absolute stability.
VNLR Stability. In Sec. 117 we saw that the basic form of the nega
tive conductance developed in the active region was
Y in = (1  a)G,
This equation is of the identical form with the negativeimpedance func
tion derived for the CNLR devices. It follows, from Eq. (1138), that
the substitution of a(<o) into the admittance term results in
Fig. 1127. Equivalent input network of
the VNLR holding during the active
region and the external load.
Jc 2 + O) 2
(1142)
Equation (1142) yields the equivalent input network given in Fig. 1127,
which we note to be simply the dual of the network of Fig. 1125.
To ensure that the poles of the controlling voltage function will not
he in the right half plane, the zeros of the admittance function must
be restricted to the left half plane. The two required conditions for
absolute system stability become
G
L <
> G x (l  a )
G x + G
GGi(a  l)<o.
(1143)
(1144)
Equations (1143) and (1144) are the duals of Eqs. (1140) and (1141)
previously derived. And as before, when neither equation is satisfied,
the load line also intersects the positiveresistance regions, thus establish
ing bistable circuit operation. Satisfaction of only Eq. (1143) allows
astable behavior for L larger than the minimum value of Eq. (1144).
346
SWITCHING
[Chap. 11
Just as with the CNLR, this circuit will also exhibit an almost sinusoidal
oscillation for small values of L.
119. Improvement in Switching Time through the Use of a Nonlinear
Load. Generally, if fast signals are to be generated, and if narrow pulses
are used for triggering, operation within the saturation region of semi
conductor devices must be avoided. The long time delay introduced
by the minoritycarrier storage precludes all but the slowest switching
intervals. In order to accomplish this restriction, we rely on nonlinear
load lines which are developed with the aid of appropriately biased diodes.
With one region forbidden us, both of the remaining circuit states
must be used in a bistable; one stable point will be situated in the cut
off zone, and the other will lie in the negativeresistance region. A simple
bistable circuit for highspeed switching, featuring this arrangement, is
shown in Fig. 1128. It makes use of a single external diode to insert
Fig. 1128. Stabilization within the negativeresistance region — bistable operation.
the necessary break in the load line, ensuring that it intersects the volt
ampere characteristic only in these two regions.
The absolute stability of the point established in the completely
dissipative cutoff region, by the diode's conduction (point A), is not
open to question. But for a true bistable, we must justify the stability
of point F, which lies on the negativeresistance portion of the char
acteristic. At this point the large load resistance R definitely satisfies
one of the two stability conditions [Eq. (1140)]. If the stray capacity
is kept low enough, or if the parameters of the CNLR satisfy the second
condition [Eq. (1141)], or an equivalent condition if the device used is
neither a pointcontact nor a pnpn transistor, then point F will also be
absolutely stable. From Eq. (1141) we see that the smaller the value
of negative resistance, the larger the capacity that can be tolerated with
out changing the stable to an unstable point.
Switching in the circuit of Fig. 1128 will always be from A to F and
from F back to A. We depend on the combination of the stray capacity
and the inductive component present in the drivingpoint impedance
of the CNLR to slow the system response enough so that the path of
Sec. 119] negativeresistance switching circuits
347
operation will never enter the saturation region (see the dashed path of
Fig. 1128).
A stableoperation Stabilization. An alternative technique for prevent
ing the transistor from being driven into saturation, illustrated in Fig.
1129, is most widely applied in astable systems. This circuit configura
tion requires the insertion of a resistor fi 2 in series with C to stabilize
the intersection occurring in the negativeresistance region. Suppose
that the charge stored in C backbiases the diode while the operating
point is at A. Then the additional resistance increases the net circuit
dissipation to a point where it will exceed the net energy supply.
In order to find the conditions which must be satisfied to stabilize
point A when the diode is backbiased, we must solve the complete
model of the system. Representing the CNLR by the model of Fig. 1 125,
il
1
.
i?>
C5
v m .
i
n
CNLR
+ **
D i
Fig. 1129. Stabilization within the negativeresistance region — astable operation.
the two conditions which will ensure absolute circuit stability in the
negativeresistance region are
R > Ri(l  «„)
p > (oco ~ l)RR\
Ri + R
Ri(l  a ) + R [Ri(l  ao) + R]Cu c
where the negative resistance is R\{1 — <*o) The first condition is the
same as expressed in Eq. (1140). From the second condition we see
that the smallest value of iE 2 that will maintain the circuit stable, regard
less of the value of C, is a resistance equal in magnitude to the parallel
combination of R and (1 — ad)Ri. This makes the net resistance across
the energystorage element positive.
However, C eventually charges, and at the instant that the voltage at
its lower terminal reaches zero, the diode again conducts. The circuit
becomes astable, and the circuit flips clockwise to point B. As discussed
in Sec. 113, the capacitor charges toward the Th6venin equivalent
voltage across its terminals, eventually reaching the peak, V p .
348
SWITCHING
[Chap. 11
In this circuit, the voltage across the terminals of the CNLR can
change instantaneously because this change will not appear across C
but will be coupled by C to R 2 . Any voltage drop, no matter how small,
backbiases the diode. The operating point thus jumps from the peak
to the now stable point A, and the diode is driven negative by this
same voltage change. C again charges, with its bottom terminal rising
from —(Vp — Va) toward V and with the approximate time constant
r 2 s (R» + R   R n )C
It eventually reaches zero, the diode conducts, the circuit again becomes
astable, and the cycle repeats. The approximate voltage waveshapes
V A
/*
/ l
[
(
1 j i
t
t
V D
(V„V A )
__
/*
[/
Fig. 1130. Voltage waveshapes appearing at the input of CNLR and across the
diode in the circuit of Fig. 1130.
generated across the diode and at the input to the CNLR are sketched
in Fig. 1130.
An arrangement similar to that shown in Fig. 1129 might also be
used in a monostable circuit. In this case, returning Ri to a small nega
tive voltage — V, instead of to a large positive one, will keep the diode
backbiased and guarantee the stability of point A. After an external
trigger forces the diode into conduction, the circuit momentarily becomes
freerunning and switches clockwise into the cutoff zone. If the circuit
parameters are properly adjusted, the capacitor charging current will
maintain diode conduction during the complete interval that the operat
ing point remains in this region.
Subsequent to reaching the peak value of the CNLR, any drop in
voltage, coupled by C, will turn off the diode and reestablish point A
as the single stable point. The operating point drops from the peak
to point A, and the circuit recovers. Since the steadystate voltage
across the diode is — Vj the diode remains backbiased and the operating
point finally stabilizes at A .
SEC. 1110] NEGATIVEBESISTANCS! SWITCHING CIBCOTTS
349
1110. Negativeimpedance Converters. The basic approach taken
in developing negativeinput characteristics, through the use of active
elements, is illustrated by the two typical block diagrams of Fig. 1131.
In the circuit of Fig. 11 3 la, when we write the input node equation,
assuming an ideal voltage amplifier,
ei — Avei
we readily arrive at the recognized form of the Miller input impedance
Zi. =
(1145)
If the voltage amplification within the active region is greater than
unity, then the input impedance becomes negative. Under the particular
»1
A»
S—
t
ei
1
f V
**c >
"
1
(a) (b)
Fia. 1131. Basic negativeimpedance converters (NIC), (a) Voltagecontrolled cir
cuit; (6) currentcontrolled configuration.
conditions where A r is set equal to +2, we obtain the following very
convenient result:
Z* = Z L (1146)
In general, Ay is a twostage amplifier having definite frequency char
acteristics and the overall response will not be in as simple a form as
Eq. (1146).
The simplest possible operation of the basic currentcontrolled circuit
of Fig. 11316 is where the complete input driving current flows through
the shortcircuited input of the ideal current amplifier. Under these
circumstances the input loop equation is
ei = (*i  i»)Z L = fi(l  A C )Z L (1147)
and by making A e = 2,
Ziu = J = — Zi
However, if only a small percentage of the input current flows directly
to the output, the load current becomes
t'i = KiL — A,
■H
350
SWITCHING
[Chap. 11
And if this term is substituted into Eq. (1147), the input impedance may
be written
Z„ = (K  A C )Z L
(1148)
where K < 1 and A c is the forward current gain.
For example, in the pnpn transistor, A c = 0. Consequently, when
these terms are substituted into Eq. (1147), the resultant expression is
identical with that given in Eq. (1131):
Z„ = (1  0)Z L
All the devices that were discussed in Sees. 116 and 117 could just
as easily have been analyzed by converting their equivalent circuits
into the appropriate block diagrams. By doing so we would have lost
the insight that was gained in examining the actual device. The major
role served by the block diagrams is to indicate the general conditions
toward which we must design if
(loti)/! we wish to develop negativeinput
characteristics.
Consider, for example, the stand
ard negative impedance converter
(NIC) circuit of Fig. 1132. Except
for the additional bias resistors and
batteries, it is of the identical con
figuration of the composite transistor
circuit used to describe the behavior
of the pnpn transistor. Once we
recognize that the output load is
Rl II R, we can also identify this cir
cuit as a practical form of the basic
impedance converter of Fig. 11316.
Only a small fraction of the input current flows directly to the output,
and we must therefore evaluate both K and A c in Eq. (1148). The
current through the direct transmission path is simply (1 — ai)i' lt and
since c*i is very close to unity, this term becomes insignificant compared
with i' x . K in Eq. (1148) may be taken as zero. Thus the input
pnp stage acts as a simple current amplifier having a gain, from the
emitter to the collector, of approximately unity.
The output current of T x divides between the collector load R and the
input impedance to T 2 of R(Pz + 1):
Fio. 1132. A practical negativeimped
ance converter — currentcontrolled oper
ation.
^2 =
t'xfl
03, + l)R + R
NEGATIVERESISTANCE SWITCHING CIRCUITS
and the current gain to the emitter of Ti becomes
A B =
^1
351
(1149)
ft + 2'
where the approximation holds for a highgain transistor.
Substituting K = and A c = 1 into Eq. (1148), we find the effective
input impedance to be
Z[ n = (R\\Rl)
But this is paralleled by the input resistance R, which cancels the nega
tive term due to — R. The only remaining term is that dependent on the
load Rl\
Z in = Rl (1150)
This same result can also be obtained by noting that the current flow
through Rl \\ R is — i[. Since a short circuit exists through the emitter
and base of T lf the voltage drop across the load must be identically
»i = ~i'i(RL II R)
By taking into account the role of the input resistor R, the input imped
ance will be as given by Eq. (1150).
Of course, here also the pure negative input resistance would exist
only at very low frequencies. At the higher frequencies, the frequency
dependence of a t and /3 2 will introduce the additional inductive component
shown in Fig. 1125.
PROBLEMS
111. A certain CNLR with the characteristics given in Fig. 1133 is used in the
basic sweep of Fig. 113. The load resistance R is 4 K, and the external timing capac
ity is 2 ni. Plot the current and voltage waveshapes after the circuit is triggered if
V n n is as given below. In each case state the area of the current impulse which must
be injected into the node to cause switching.
(O) Vnn = 40 volts.
(b) V„„ = 40 volts.
112. (a) What limits of load resistance
will permit bistable operation of the
CNLR of Fig. 1133?
(6) The load resistance used with CNLR
of Fig. 1133 is 10 K. What ranges of
V„„ permit operation as a monostable
circuit ?
(c) Repeat part o if the load resistance
is now 2 K.
113. A 5jif capacitor is connected directly across the input terminals of the CNLR,
which has the characteristics of Fig. 1133. Sketch the current and voltage wave
shapes, labeling them with respect to time constants, voltage and current values, and
times.
Fig. 1133
352
SWITCHING
[Chap. 11
114. At t = the network N of Fig. 1134 is activated by the voltage impulse
indicated. Sketch i n and v» as functions of time, labeling clearly all break points and
time constants.
116. (a) Give the range of external resistance that will make the circuit of Fig.
1134 astable when the battery voltage is between the limits of 4 < F„„ < 10 volts.
(6) Sketch the output waveshapes, labeling them completely, when B = 10 ohms
and V„ n = 10 volts.
2S(t)
Fig. 1134
116. Calculate and plot the CNLR characteristics seen when looking across C in
the monostable circuit of Fig. 938 (Prob. 914).
117. Plot the CNLR characteristics for the circuit whose values are given in Fig.
1112 when R = 100 K and flj = 25 K. All other parameters remain as before.
Superimpose this plot on a copy of Fig. 1113.
118. (a) Evaluate and plot the transfer characteristic, i.e., collector voltage of T t
versus i a , for the monostable multivibrator (Fig. 1112) discussed in Example 112.
(6) Sketch one complete cycle of i a after the circuit is triggered. Using the curve
found in part o, draw to scale 1 cycle of e C 2.
119. Plot the voltampere characteristic for the multivibrator of Fig. 1112 when
R> is increased to 100 K. Compare your results with Fig. 1113. Which regions are
most affected by the increase in i? 3 ? What does it do to the modes of operation?
1110. Calculate and plot the voltampere characteristic seen across the inductance
in the circuit of Fig. 1135. Using this plot, sketch and label the inductive current ii
after a trigger is applied.
+20 v
/360 <100
1135
1111. The pointcontact transistor of Fig. 1114 has a = 1.2, n = 200 ohms, and
r„ = 150 K. Assume that the peak point of the input negativeresistance character
istic is located at the origin. Specify all circuit parameters necessary to set the valley
NEGATIVERESISTANCE SWITCHING CIRCUITS
353
point at —10 volts and 10 ma. Sketch the driving and transfer characteristics,
labeling all slopes.
1112. Plot the magnitude and phase of the input impedance of the pointcontact
transistor in its active region as a function of a. iJi = 5 K, « = 1.5, and u c = 10*.
1113. Assume that the pnpn transistor of Fig. 1117 is composed of two indi
vidual units, each having a = 0.98. In all other respects these transistors are ideal
units. Plot the input voltampere and the ««2 versus ii characteristics of this device.
Specify the break points when Ri = Ri = 1 K and E = 10 volts. Make all reason
able approximations in your calculations.
1114. The unijunction transistor shown in Fig. ll36o may be represented
approximately by the voltampere characteristic of Fig. 11366. Sketch the wave
forms of V. and /, as functions of time, labeling clearly all break points and time
constants.
l/rf=T=
■=LE kh
via ni
Fig. 1136
1115. The unijunction diode of Fig. 1118 has the following parameters: rn =
140, r« = 60, r, = 30, y = 2.5, and R, = 10 ohms. The backbiased input imped
ance is 50 K.
(a) Plot the input voltampere characteristics when Ebb = 20 volts.
(6) Repeat part a when a 2,000ohm resistor is inserted in series with 6 S and a
1,000ohm resistance in series with 6i.
1116. The pointcontact transistor of Prob. 1111 is used in the configuration of
Fig. 1123. R } = ijj = 5 K, and the reverse resistance of both the collector and base
is 500 K.
(a) Find E cc = E„ necessary to produce a current swing of 2 ma when the circuit
operates as an astable device.
(6) The external conductance is twice the limiting value. What is the smallest
battery in series with the inductance that will allow the circuit to freerun? What
is the largest series battery? v
(c) If the series inductance is 100 mh, what is the smallest cutoff frequency of the
transistor for satisfactory operation? Where are the poles and zeros of Y(p) located
under this condition?
1117. (a) Prove the equivalency of the drivingpoint network of Fig. 1125.
(6) Derive Eq. (1139) and verify Eqs. (1140) and (1141).
(c) Show that as L —> 0, the stability condition is satisfied by Eq. (116).
1118. (a) The tunnel diode of Fig. ll37a is connected to an external circuit
consisting of a 10mh inductance, a 1ohm load resistance, and a 250mv bias source.
Approximate the characteristic by three line segments, and plot the terminalvoltage
waveshape.
(b) If R, = 0.3 ohm, L, = 2 pii, and C, = 50 wit in the activeregion model of Fig.
1122, what is the highest frequency of operation?
354
SWITCHING
[Chap. 11
(c) Express the absolute stability conditions for the general tunneldiode model of
Fig. 11226.
1119. A tunnel diode having the characteristic of Fig. ll37a is used as a coinci
dence gate. The circuit appears in Fig. ll37b. Positive input pulses applied at e%
and ea have an amplitude of 1.5 volts and a duration of 0.1 iiaec. Plot the output
voltage under the following conditions:
(o) Eu, = 0, «i or e% present.
(6) En, = 0, «i and ei simultaneously applied.
(c) Ebb = 250 mv, ei or e 2 present.
(d) Ebb = 250 mv, ei or ei simultaneously applied.
,
k
30

20
>
10
— >•
J 61 0.2 0.3 0.4 0.5 0.6 0.7 0.8
v d , volts
(b)
(a)
Fig. 1137
1120. Figure ll38a shows a fivesegmented approximation to a tunnel diode's
voltampere characteristic. This particular device is employed in the switching cir
KS(t)
10 mh
:5oo
t
(b)
Fig. 1138
NEGATIVERESISTANCE SWITCHING CIRCUITS
355
cuit of Fig. 11386. A voltage impulse having an area K of 40 X 10~* voltsec is used
for triggering. Sketch and label the output voltage for both positive and negative
impulses under the following conditions:
(a) JS» = 0.
(fc) En  1 volt.
1121. A train of 1volt positive and negative pulses, 0.1 /isec wide and spaced
2 /isec apart, is applied to the circuit of Fig. 1139. D\ and D 2 are tunnel diodes whose
voltampere characteristic is linearized in the manner shown. Da is a fastacting
decoupling diode and may be considered ideal for the purposes of this problem.
(o) Sketch and label the output voltage over a 10/isec interval.
(6) Repeat part a when D» is removed and the coupling is directly through the
100ohm resistor.
200 mv
100/ih;
?lv
•IK
Input
pulse o — VvV
train 300
Wv
100
200 <
M
3CA,
Fio. 1139
1122. The terminal characteristics of a certain CNLR are shown in Fig. 1140.
Also shown is the equivalent circuit for the device in its active region.
NLR (static)
Active region
Fig. 1140
External circuit
(a) When the external network is connected by closing switches at t = 0, deter
mine in as a function of time. (Hint: Solve for the poles of the network in its active
region and also for the steadystate component of i„; then write i n = itrtmient + »'.» and
evaluate the necessary arbitrary constant.)
(b) How does the circuit function if C is reduced by a factor of 2? If it is increased
by a factor of 10? Give a qualitative answer and describe the expected terminal
waveshape.
1123. (a) Repeat Prob. 1114 when the 1^if timing element is shunted by a Zener
diode which conducts for V. > 8 volts (Fig. 1128).
(6) To what value must C be reduced before point F becomes the second stable
point?
356 switching [Chap. 11
1124. Prove that the insertion of a resistor in series with C (as in Fig. 1129) can
make a normally unstable circuit stable. [Hint: Return to the drivingpoint net
work of Fig. 1125 and find the conditions necessary to restrict the zeros of Z(p) to
the lefthalf plane when iJj is in series with C. Compare these with the stability
conditions given by Eqs. (1140) and (1141). Equation (1140) is satisfied and
(1141) is not when R t is removed.]
1126. In the negativeimpedance converter of Fig. 1132, Bl ■= B = 10 K,
Vi = Vs = 10 volts, and 3i = /Ss = 50. Calculate the input voltampere character
istic. Specify the slopes and intercepts of each line segment. Find the current
transfer characteristics from the input to the load, Rl.
BIBLIOGRAPHY
Anderson, A. E.: Transistors in Switching Circuits, Proc. I BE, vol. 40, no. 11, pp.
15411548, 1952.
Beale, I. E. A., W. L. Stephenson, and E. Wolfendale: A Study of High Speed Ava
lanche Transistors, Proc. IEE {London), pt. B, vol. 104, pp. 394402, July, 1957.
Ebers, J. J.: Fourterminal pnpn Transistors, Proc. I BE, vol. 40, no. 11, pp. 1361
1365, 1952.
Esaki, L.: Letter to the Editor, Phys. Rev., vol. 109, pp. 603604, Jan. 15, 1958.
Farley, B. G.: Dynamics of Transistor Negative Resistance Circuits, Proc. IBE,
vol. 40, no. 11, 14971508, 1952.
Hall, R. N.: Tunnel Diodes, IBE Trans, on Electron Devices, vol. ED7, pp. 19,
January, 1960.
Leak, I. A., and V. P. Mathis: The Doublebase Diode: A New Semiconducting
Device, IRE Conv. Record, pt. 6, p. 2, 1953.
Linvill, J. G. : Transistor Negative Impedance Converters, Proc. IRE, vol. 41, no. 6,
pp. 725729, 1953.
Lo, A. W. : Transistor Trigger Circuits, Proc. IRE, vol. 40, no. 11, pp. 15311541, 1952.
Merrill, J. L.: Theory of the Negative Impedance Converter, Bell System Tech. J.,
vol. 30, no. 1, pp. 88109, 1951.
Shea, R. F.: "Transistor Circuit Engineering," John Wiley & Sons, Inc., New York,
1957.
Shockley, W., and J. F. Gibbons: Introduction to the Fourlayer Diode, Semiconduc
tor Prods., JanuaryFebruary, 1958, pp. 913.
Suran, J. J., and E. Keonjian: A Semiconductor Diode Multivibrator, Proc. IRE,
vol. 43, no. 7, pp. 814820, 1955.
CHAPTER 12
THE BLOCKING OSCILLATOR
Most of the multivibrators discussed in Chaps. 9, 10, and 11 depended
for their timing on the energystorage element maintaining the tube or
transistor off for the required interval. Because an active element is
essentially an open circuit within its cutoff region, the output pulse is
available only at a relatively high impedance level and with a low power
content. Furthermore, stray capacity slows the fast rising and falling
edges, preventing the generation of extremely narrow pulses. One
method of overcoming these disadvantages is to time the pulse within
the highcurrent lowimpedance saturation region.
The blocking oscillator is one circuit which is so designed. Only
a single active element is necessary, with a specially designed transformer
used for timing as well as for phasing the regeneration. Because this
deceptively simple circuit is so widely used, in computers, radar, tele
vision, etc., it will be treated in some detail. In doing so, we extend the
approximation methods of analysis to multipleenergystorageelement
systems. We shall also be obliged to consider the problems posed when
some of the passive components exhibit nonlinearity. No attempt will
be made to obtain an exact solution, but the analysis will adequately
explain the role of the various elements and the functioning of the circuit.
121. Some Introductory Remarks. Figure 121 shows one possible
form of the blocking oscillator, a collectoremittercoupled circuit,
together with its general piecewiselinear representation. As possible
alternative configurations, the transformer may couple either the collector
or the emitter to the base. In brief, the sequence of monostable opera
tion is the following. The transistor is initially backbiased, and in
response to an externally applied trigger, it is forced into the active
region. Provided that the transformer has the proper turns ratio and
connections (introducing a sign change), the net loop gain will be posi
tive and greater than unity. Regeneration drives the circuit into
saturation. The transistor acts as a switch, which operates with a
minimum amount of input energy and which connects Em, across the
primary of the transformer.
The full duration of the saturation interval is controlled, as is all
357
358
SWITCHING
[Chap. 12
timing, by the energystorage elements present (in this circuit by the
magnetizing inductance L m and the coupling capacitor C). When gener
ating long pulses, an evaluation of the width may be further complicated
if the large current buildup in the magnetizing inductance L m drives
the transformer into saturation. For short pulses, the timing will be
influenced by the rate of flux penetration into the core (Sees. 123, 132,
and 133). The solution of the complete problem is quite complicated,
involving as it does multiple modes of energy storage, core properties,
and hysteresis.
Ideal transformer
R» r u , R L » r n
Fig. 121. Collectoremittercoupled blocking oscillator and complete piecewiselinear
model.
Eventually the circuit reenters the regenerative region and switches
back off. The oscillator recovers and remains off until the next trigger
is applied.
The blockingoscillator transformer is a very important component,
so critical, in fact, that the circuit is designed about its characteristics.
By interleaving the coil windings and by using a highpermeability core
material, the leakage inductance and stray capacity are minimized,
and consequently so is the rise time. For example, a transformer
designed to generate a 1itsec pulse may have a nominal magnetizing
inductance of 1.0 mh compared with a leakage inductance of only 20 to
50 ^h. As it is quite small physically, the stray capacity would only be
10 wi .
The transformer usually includes one or more tertiary windings which
are used to couple the generated pulse to the isolated external circuit.
Sec. 122] the blocking oscillator 359
122. An Inductively Timed Blocking Oscillator. The following sim
plifications are now possible:
1. The input impedance of the transistor in the active and saturation
regions is so very low that C, will have a negligible effect on the rise time.
2. The coupling capacitor C is very large, and we may therefore assume
that its terminal voltage remains constant over the entire pulse interval.
3. Both R and Rl are much greater than rn
4. The transformer core never saturates.
Using the above assumptions, the incremental model holding during
the active region reduces to the one shown in Fig. 122o. The two node
equations are
(7 c + i) e °i ei = °*« (12  lo)
 L Cc + (~ + i~) «i = (1216)
Substituting i. = — ei/nru, the solution of these equations yields the
system's pole, which is located at
p^_ r,(lan) (122)
From Eq. (122), we conclude that the condition which must be satisfied
if the circuit is to be regenerative (a pole located in the right half plane) is
an > 1
and that this corresponds to a loop current gain greater than unity.
Before the circuit enters into its active region, the value of the collector
voltage is Em,; immediately after the pulse turns the transistor on, the
collector drops to zero. The full supply voltage appears across L e , and
the transistor saturates rapidly at an extremely low current. To find
the time required, we must know the starting point of the exponential
buildup. But this depends on the energy content of the excitation
pulse, and since, in general, it is unknown, the problem is not completely
defined. For a rough order of magnitude, we can approximate the
initial switching time by 2.2 time constants of the positive exponential of
Eq. (122).
f i S 2.2 7 ^ (123)
{an — l)r c
With a good pulse transformer Eq. (123) may even yield a time of less
than 10 8 sec. The turnon time of the transistor may be very much
longer, and if it is, it will predominate — the solution of Eq. (123) will
be totally without meaning.
Once the transistor saturates, the models of Fig. 122& and c will be
used to define the collector and emitter current buildup. Because
360 switching [Chap. 12
L. «C L m , the actual doubleenergy problem will be treated by the meth
ods of Chap. 1, i.e., by assuming that the current buildup in L, is com
plete before that of L m starts. The waveshapes obtained from this
approximation are sketched in Fig. 124.
Fig. 122. Models holding for the various regions of blockingoscillator operation —
timing due to the transformer's magnetizing inductance. Waveshapes appear in
Fig. 124. (a) Activeregion incremental model; (6) models for the initial portion of
the saturation region; (c) models for the final portion of the saturation region; (d)
model for the recovery region.
The primary voltage exponentially approaches a peak value of — En,
with a time constant due to the leakage inductance and the input resist
ance of the transistor reflected through the ideal transformer.
T2
L«
where ru is the saturation value of the input resistance.
THE BLOCKING OSCILLATOR
361
Sec. 122]
Since the voltage across C remains constant at E a , the net drop across
rn will reach a peak of
Eg
n
Eem — E a
and the corresponding value of emitter current is given by
E,
= ±(^E a ) (124)
r n \n )
For saturation I m > 0, and therefore the circuit must be designed so that
em ~~
E ^>E a
n
If this condition is not satisfied, the transistor will immediately switch
back into its active region and no pulse will be generated.
v
«/,,
hi ■:
C/< >hm
le***ei
0' .
i«0
Fig. 123. Transistor collector voltampere characteristics — groundedbase connection
— showing the path of operation of the blocking oscillator. The transition time
between the various points is indicated.
The primary current, which flows into the collector, is related to the
emitter current by the transformer turns ratio n.
Iim — — — 
(125)
This segment of the path of operation lies between points B and C in
Fig. 123. The time required for the rise; from 10 to 90 per cent of the
final value, as defined in Eq. (132), is
h = 2.2r j = 2.2 
(126)
Since ru is quite small, in order to minimize the rise time, L, must also be
small.
Next the magnetizing current starts building up. If the small resist
ances of the conducting collector diode and the transformer primary
362 switching [Chap. 12
are neglected, the model reduces to the one shown in Fig. 122c. The
constant voltage across L m produces a linearly increasing magnetizing
current
£>« (127)
which increases the total collector current, driving the operating point from
C toward D in Fig. 123. When i c finally reaches al em , the net collector
diode current drops to zero and the transistor reenters its active region.
Since the emitter current remains almost constant over the range of
interest, from Eqs. (124), (125), and (127), we obtain
ic = Ji. + im = — (—  E^\ + ^5 t (12 8)
run \n ) L m v '
By setting i c equal to ale„, we find that the final portion of the pulse
lasts for
hL, na ~
«_=ll(l_*KA (12 . 9)
n 2 ru \ E»J v '
Because h must be real for any combination of terms, Eq. (129) again
proves that the conditions which must be satisfied for the proper circuit
operation are na > 1 and nE a < E bb . In this mode, the segment
defined by the linear current buildup in the magnetizing inductance
constitutes the major portion of the desired pulse (that is, W^>ti) and
Eq. (129) essentially defines the complete pulse width.
The transistor switches back off in much the same manner as it turned
on, and the circuit begins its recovery. As the current in L„ decays, the
change in the direction of di m /dt produces a large backswing of voltage
which, coupled by the transformer, aids in backbiasing the emitter.
From the model of Fig. 122d, we see that the primary voltage would
be given by the product of the peak value of the magnetizing current and
the reflected resistance.
E mm = I mm 7i>R = (ocI em  I lm )n*R (1210)
The collector voltage jumps from zero to
E cm = Etb f E mm
with the emitter voltage going to
Em
mm
E em = E„ +
n
Recovery toward the initial conditions is with the fast time constant
'•  k < 12  n)
Sec. 122] the blocking oscillator 363
Instead of the current and voltage decaying monotonically toward
the steadystate values, the magnetizing inductance may resonate with
the stray capacity of the circuit, producing ringing in the output. The
initial amplitude depends on the amount of energy previously stored in
al a
t f
*m i c Am
196 ma
2.22*! 16.26 //sec
i i
200' m'a
fct
2.22
'.) 16.26.Msec
587 v
21 vl
18.48 ^sec
191 v
V*
E a
3v
V
18.48 /tsec
t
E»
At
W_
n
E a 
Piq. 124. Waveshapes produced in the inductively timed blocking oscillator. The
values given are those found in Example 121.
the transformer. If the oscillations damp out slowly, the next change in
polarity may force the transistor back into its active region, resulting in a
second, undesired output pulse — the circuit would become astable.
Example 121. The blocking oscillator of Fig. 121 employs a transformer having
L m = 2.5 mh, L. •» 100 fth, » = 3, and C, = 20 npi. The transistor is charac
terized by
Tix = 20 ohms
r c = 1 megohm
or  0.98
turnon time 0.1 /usee
turnoff time 0. 1 ^sec
364 switching [Chap. 12
And the remaining parameters are E bb = 21 volts, E a = 3 volts, R = 500 ohms,
Rl = 5 K, and C = 10 /if . We wish to construct the various waveshapes encountered
(Fig. 124).
From Eq. (123) the saturation time is given by
*' = 22 (0.98 X 3  1)10"  10 ~ Se °
but since the turnon time is 0. 1 ^sec, this would be the proper value of h; the equation
gives a solution that is a factor of 10~ 3 too small.
The emitter current next rises to
I.m = Ho( 2 ^  3) = 200 ma
with the time constant t 2 = 0.556 jusec. At the end of this interval the collector
current is only
, 200 ma „_
lim = g — = 67 ma
In the saturation region the magnetizing current starts increasing as
20 ,
2.5 mh
When this current builds up to the difference between <*/,„ and /,„, the transistor
turns off. Under the conditions of this problem, the transformer must tolerate
129 ma of magnetizing current without saturating. Thus t, is given by
[0.98(200)  67J10' = ( X 10') J
or t% = 16.26 iiaec.
The peak voltage developed across the primary is
Eim  (196  67)10» X 9 X 500  566 volts
Unless limited by the judicious use of diodes, as shown in Fig. 125, this extremely
large voltage will destroy the transistor. The voltage at the emitter winding is only
Eim/n, or
E lm = 56 % = 188 volts
Recovery is with the relatively fast time constant
2.5 X 10«
9 X 500
= 0.54 /isec
Figure 125 shows how to prevent the saturation of the transistor as
well as how to limit the excessive amplitude backswing. Zener diode
D s fires before the transistor can saturate and so sets a bottoming voltage
of Es volts. The saturation conditions are transferred from the collector
diode to the Zener diode. Except that the primary voltage builds up
to Eu, — E a volts instead of to E M , the solution of the circuit is identical
with that previously discussed. Diode Db conducts on the backswing
if the terminal voltage exceeds E B . Since it shunts the coil with its
Sec. 123]
THE BLOCKING OSCILLATOR
365
low forward resistance, the discharge time constant would be greatly
increased. In the solution of the recovery interval, two time constants
must be considered, the long one holding while Db conducts and the fast
one (Eq. 1211) holding after the voltage decays below Eb.
t 3
Fig. 125. Zenerdiode limiting of the collector voltage of the blocking oscillator.
123. Transformer Core Properties and Saturation. The inductance
of the blockingoscillator transformer is proportional to the ratio of flux
linkages and the current establishing this flux.
N ^ 10 8
at
henrys
(1212)
At low frequencies or even at dc, the relationship between the flux
and the excitation is adequately expressed by the hysteresis curve (Fig.
126). Assuming ideal conditions, B is simply the flux per unit area and
H is proportional to the ampereturns per unit length. For large signals,
B = pH, and hence the inductance is proportional to the permeability
of the core. From the hysteresis loop of Fig. 126o, we can see that ix
varies over rather wide limits as the core flux increases from zero to
saturation.
If the hysteresis loop is approximated by the rectangular plot of Fig.
126b, we avoid the necessity of treating a circuit containing a continu
ously changing inductance. After lumping the core losses together with
the external resistance, it can be assumed that
and
L=L m
M < N
Itl > \i.\
366
SWITCHING
[Chap. 12
where i, is the value of the coil current corresponding to the saturation
magnetizing force H,. The inductance in the saturation region approxi
mates that of the coil in air; since L, <£. L m , it is often taken as zero.
Because it requires finite time for the applied excitation field H to
propagate into the core and establish and align the randomly oriented
magnetic domains, the above formulation is not completely correct.
With a pulse excitation, the flux is initially only at the surface, and con
sequently the effective permeability is quite low. It increases as the field
penetrates into the core. Even if the excitation is well above the satura
tion value, it may still take several microseconds to saturate a thin core.
The penetration time may be long compared with the width of the pulse
generated in the blocking oscillator, and consequently the assumption
of a constant L m is not valid.
(a) (b)
Fig. 126. Hysteresis curve and piecewiselinear representation.
In order to avoid this inconsistency, we would have to postulate a
magnetizing inductance whose value increases with time from slightly
above that of an aircore coil to a peak given by the steadystate core
characteristics. A circuit element which exhibits a response of this
nature is impossible to use in any simple approach to circuit analysis. We
shall therefore continue to explain the blockingoscillator operation on the
basis of a fixed L m and accept the resulting errors. However, the accu
racy of calculations can be improved if the inductance value used is not
that at direct current but rather that measured under pulsed conditions.
To summarize:
1. For pulse durations greater than several microseconds, the assump
tion of a constant L m gives reasonably correot times.
2. When generating pulses of less than 1 or 2 ^sec, the duration cannot
be accurately calculated from any simple model. If a constant L„ is
assumed, then the actual pulse will terminate sooner than calculated.
3. A finite time is required to saturate the core, and therefore this
problem need be considered only when generating wide pulses.
Returning to the blocking oscillator, we shall now examine the opera
tion when the core does saturate. Up to this point, the response will be
Sec. 124] the blocking oscillator 367
identical with that discussed in Sec. 122. Since point C of Fig. 123
indicated the start of the magnetizingcurrent buildup, the saturation
current point D' would fall somewhere along line segment CD; for a
highpermeability core, it might be quite close to the initial point. This
is the termination point of the pulse; the lower the saturation value, the
more it is foreshortened.
Once the transformer saturates, L m drops to a very small value. The
primary current increases rapidly, limited only by the total series resist
ance. At the same time, the secondary voltage falls, the emitter current
drops, and the transistor again becomes active. As the stored energy
dissipates, the lefthand branch of the hysteresis curve is traversed by the
operating point. As a result, a large reverse voltage pulse appears at
the collector. The time required for the decay depends on the nature
of the transformer core. (This problem is treated in more detail in Chap.
13.) Because of the extremely nonlinear behavior, it is difficult to define
the decay time in terms of the specific circuit parameters. In general,
the manufacturer's specifications as to the rise and fall times and pulse
width would be used in the circuit design.
124. Capacitively Timed Blocking Oscillator. The coupling capac
itor C is now reduced in size until it alone will control the pulse dura
tion. Instead of the increasing collector current bringing the transistor
out of saturation, the decreasing emitter current, due to the charging
of C, will now so serve. To simplify the following analysis, we shall
assume, quite incorrectly, that the magnetizing current remains zero
over the complete pulse interval. Moreover, since the On time constant
is on the same order of magnitude as LJnhw, the approximation that the
initial buildup is complete before the timing element starts charging
is no longer completely valid. We shall, however, assume that the peak
values expressed in Eqs. (124) and (125) are still correct and shall
begin our analysis from this point. By doing so, the nature of the
reduction in the pulse duration may be seen most readily.
From the model of Fig. 127o, we can see that C charges from its
initial value of E a toward the TheVenin equivalent voltage
The time constant is
r 3 = C(R  r„) » Cm
During this interval, the emitter current decays from its peak[Eq. (124)]
I em — ( E a ]
m\n )
368 SWITCHING
toward the steadystate value of
E.
[Chap. 12
/.(«>) = 
r« + R — R
The collector current will always be 1/n times as large. Consequently,
because the transistor starts out saturated, it will remain so until I,
falls to zero.
Fig. 127. Models holding for the capacitively timed blocking oscillator, (a) Model
holding during the charging interval; (6) model defining the recovery period.
The duration of the unstable state, as found from the exponential
charging equation, is given by
T3
In
r.(«>)  I.
or by substituting the appropriate terms,
^^[^(iF. 1 )] (12 " 14)
If R 5J> rn, this portion of the pulse will consist of virtually the complete
exponential. To avoid the ambiguity arising when the switching point
is close to the steadystate value, E a might be made highly negative and
R might be reduced.
At the termination of the pulse, the transistor switches off and the
recovery model reduces to the one shown in Fig. 1276. Because the
approximations made in the course of the above analysis neglected the
energy storage in the transformer, the exact nature of the response over
this portion of the cycle is somewhat difficult to define. One type of
response would be expected when the secondary magnetizing inductance
Sec. 124]
THE BLOCKING OSCILLATOB
369
limits the rate of capacitor current flow, and yet another type where the
transformer saturates rapidly. Regardless of the particular controlling
element, the energy storage will almost always produce a large spike of
voltage, and then either ringing (which must be suppressed) or an
exponential decay back toward the original state. Waveshapes appear
ing at all elements, illustrating the type of recovery to be expected,
appear in Fig. 128.
f*3.8,«sec»i
t
t
_L_jw
1 i/v
\ V
t
Iem r U
_____\J
Fig. 128. Waveshapes appearing in the capacitively timed blocking oscillator. The
recovery region illustrates ringing.
We shall now consider the change in the response of the blocking
oscillator of Example 121 when C is reduced from 10 to 0.02 /if. From
Eq. (1214), the duration of the final portion of transistor saturation
lasts for
h = Cn
■[
1 + W(7
t 40 V
1)
Since this is almost the complete exponential,
; 4CVxi = 4(0.02 X 10o) X 20 = 1.6 M sec
370
SWITCHING
[Chap. 12
K»m
The pulse width, under this condition, is approximately onetenth the
time calculated when L m controlled the timing.
We should note that the maximum pulse duration is due to the current
buildup in the magnetizing inductance. Either the charging of C or
the saturation of the core can only reduce this time. Furthermore, the
simultaneous magneticflux buildup means that the peak currents that
are reached and the actual pulse duration will always be less than calcu
lated on the basis of the idealizations made above. C is almost never
used to control the pulse width, but it is used to time the interval between
pulses in a freerunning blocking oscillator (Sec. 125).
125. Astable Operation. As might be anticipated from the previous
discussion of monostable and astable operation in Chaps. 10 and 11,
shifting the quiescent operating point
from the cutoff to the active region will
make the blocking oscillator freerunning.
For example, the astable circuit of Fig.
129, except for the reversal of the
emitter battery, is identical with the
monostable circuit of Fig. 121.
■ During the pulse interval, the large
emitter current charges C slightly posi
tive. After the circuit switches off,
then the accumulated charge maintains
the emitter backbiased. Eventually
the voltage from emitter to ground de
cays to zero and the circuit generates another pulse.
At the instant of switching on, the net charge in C must be zero. The
voltage at the transformer secondary has not as yet built up to E^/n,
and therefore the emitter state depends solely on the voltage across C.
If this voltage were positive, the emitter would be backbiased, and
if it were negative, the circuit would have switched on earlier. Except
for the change in the initial condition, the operation of the blocking
oscillator, during the pulse interval, is identical with that discussed in
Sees. 122 and 123. The peak negative voltage across the emitter and
the peak current flow are now given by
*»=?* I = J^ (1215)
For the purposes of computation we shall assume that this current
remains constant over the complete pulse interval. Since 7 em flows
through C, it will produce a net voltage change of
Fig. 129. An astable blocking os
cillator — voltages shown for the
saturated region.
A£ = q I.,
M
(1216)
Sec. 125]
THE BLOCKING OSCILLATOR
371
where At is the width of the pulse [At = f s + h as defined in Eqs. (126)
and (129)]. The capacitor thus charges linearly from its initial value
of zero to
J5«(A0 = +AJS .
as shown in Fig. 1210. In order to sustain a constant emitter current
over the pulse interval, the change in the capacitor voltage must be small
in comparison with the voltage across the transformer secondary.
#„»« ;
(1217)
Combining Eqs. (1215), (1216), and the inequality of Eq. (1217), we
can express one of the limiting conditions as
C»
(1218)
At the end of the pulse interval the transistor is driven hard off by
the inductive backswing. After the initial recovery with the fast time
Fig. 1210. Emitter and capacitor voltage waveshapes in the astable blocking oscillator.
constant, r 6 = L m /n*R, the charge in C will maintain the cutoff condition.
As it discharges toward — E a , with the relatively long time constant
r = RC, the emitter voltage decays from +E gm to zero. Thus the dura
tion of the interval between adjacent pulses is given by
RC\n
Eg — E ql
~E a
(1219)
And, for a given period, Eq. (1219) determines the size of R.
The astable blocking oscillator is commonly used to generate trains
of narrow pulses having relatively wide separation. The time defined
372 switching [Chap. 12
in Eq. (1219) very nearly represents the complete period of the generated
signal.
As with all freerunning circuits, the blocking oscillator may be
synchronized to an external control signal. It must, of course, have
a frequency and amplitude placing it within the regions of synchroniza
tion. Because the pulse is so very narrow when compared with the full
period, the solution of the synchronization problem would be much
closer to that found for the sawtooth sweep in Chap. 5 than to any regions
which might be constructed for a multivibrator.
Example 122. In this problem we wish to calculate the values of C and R neces
sary to establish a pulse spacing of 2.5 msec. The blocking oscillator generates a
5tiaec pulse. The significant parameters for the recovery interval are r n =40 ohms,
En,  20 volts, E„ = 1 volt, n = 2, and L m = 2.5 mh.
From the above parameters, Eq. (1218) yields
C» 5X 4 *°~' = 0.125 „f
In order to keep its size within bounds, we shall choose C = 2 /if . The peak voltage
developed across it is
E, m = ± — At = 0.625 volt
C nru
The remaining circuit component R is now found from Eq. (1219).
R ' C ** + * = 2X 2 l 5 0^ln "l' 62 5 " 2 ' 580ohm "
Ea
As a final step we should verify that the voltage backswing due to the energy stored
in the transformer will have damped out long before the transistor turns back on.
This time constant is
L„ 2.5 X 10' _ n „, 9 „,
Tt = JfiE " 4 X 2.58 X 10' " ° 242 " 8e0
and thus the backswing occupies an insignificant portion of the recovery interval. If
a Zener diode was included, this might no longer be true.
126. A Vacuumtube Blocking Oscillator. The circuit of Fig. 1211
is probably the most difficult to analyze of all those used for pulse
generation. Many words have been written purporting to detail its
operation, but none of the papers or texts have been completely success
ful. The difficulties faced are many:
1. More than a single mode of energy storage is always operative.
2. The various parasitics must be included; they establish the switch
ing time and thus place a lower limit on the pulse duration.
3. The tube operates in a region not previously defined, where its
parameters change drastically from the smallsignal operation. In this
region the values of fi, r p , and r e are not even well known.
Sec. 126]
THE BLOCKING OSCILLATOB
373
4. Because of the flux buildup and even possible core saturation,
the timing inductance becomes a function of the magnetizing current
and the pulse width (Sec. 123).
Difficult problems always have a peculiar fascination. In discussing
them, one tends to go deeper and deeper until perspective is lost and
the subject appears overly important. In an effort to avoid this pitfall,
the following arguments are concerned with two rather limited goals:
1. The development of a tube model for the new region of operation
2. The construction of a set of reasonably consistent models and the
extraction of some essential information when these models are inade
quately defined
100 K
Ideal transformer
20 v
Fig. 1211. A vacuumtube monostable blocking oscillator and its general equivalent
model — timing controlled solely by the transformer. Waveshapes produced are
shown in Fig. 1216.
Whenever the analysis becomes overly complicated or does not give
promise of rewards commensurate with the work involved, it will immedi
ately be dropped.
We shall assume that the reader has some knowledge of the waveshape
produced (Fig. 1216). Such an assumption is not unreasonable; the
laboratory must always be a close adjunct to the paper work, especially
when treating a complicated circuit. It is far easier to clear up fine
points with a little laboratory work than from a model that is at best only
an idealization.
The first step in the analysis of the vacuumtube blocking oscillator
is to define a new tube model. This becomes necessary because as the
tube in the circuit of Fig. 1213 switches on, the large plate drop, coupled
by the transformer, drives the grid beyond the point at which the previ
ously derived models hold. In fact, within the timing region, the grid
voltage may even exceed that at the plate. When this happens, the grid
loses control over the plate current. Any increase in the grid voltage
increases the total cathode current up to the emission capabilities of the
tube. But because the grid is at the highest potential, almost all the
374 switching [Chap. 12
additional current flows to it; the plate current remains essentially
constant. As can be seen in Fig. 1212a and b, when E e > Eb, the fam
ily of plate voltampere characteristics degenerates into a single curve.
It follows directly that the platecircuit response may be represented by a
simple diode having the forward resistance r p , (Fig. 1213).
200
175
150
1 125
•^100
o
* 75
50
25
X
tP 
A
r
S*
■>
/
/'
^
i.'JPn
ro
</
/'
^» ■*'£•
V2.0
1
's
' ^
115
ZS
+5
*
<&
~x
=»^
"S"
—40
50
100 150
E b , volts
200
250
*uu
175
y
*A
i
/
.
.&>
.•W
150
125
^100
75
A
 —
— —
.— *"'
—+2
5
n
—
***
B* +w
"•V
—— +
.0
.+5
*>
50
25
LBf ■
£
A
50
£ w 50
(6)
100 150
e b , volts
200
250
Fig. 1212. Vacuumtube positive grid characteristics and piecewiselinear represen
tation (after 12AU7 characteristics).
The value of the plate resistance in the absolute saturation region of
Fig. 12126 is given by
Eh\ E c \
r " hi hi
(1220)
But the plate voltage intercept of the extended activeregion character
Sec. 126] the blocking oscillator 375
istics through the point (En, hi) is
Em — —pEci
Thus, from the geometry, the plate resistance in the active region may be
expressed as
r, = *Lfl*?  <*+»*' (1221)
Comparing Eqs. (1220) and (1221) allows us to write the saturation
resistance in terms of the activeregion parameters.
rm = ^J (1222)
The grid resistance used in the model of Fig. 1213 must also be
reduced to onehalf or onequarter of the value measured at the lower
values of grid voltage. Furthermore,
.... 8° I i op
this resistance is extremely nonlinear.
At very high values of voltage and
current, the grid drivingpoint charac
teristic may even begin to approximate ■ ._
a constant voltage drop more closely Aft
than a pure resistance. In the in FlG  12 " 13  Triode model for the abso
terests of simplicity, however, we shall lute saturation Te & oa 
assume a constant resistance whose value would be found in the vicinity
of e„ S e b .
For the purposes of discussion, the operation of the vacuumtube
blocking oscillator of Fig. 1211 may be divided into the same four
regions found for the transistor circuit. These are characterized by:
I. Active region. The positive loop gain drives the tube into "abso
lute" saturation (that is, e c > e^).
II. Tube saturation region. Here the circuit voltages buifd up toward
their peak values.
III. Timing. Current buildup in the transformer, core saturation,
and/or the charging of C return the tube to the active region and ter
minate the pulse.
IV. Switching and recovery. The regeneration of the circuit turns
the tube off, and the energy previously stored must now be dissipated.
The output consists of a large voltage overshoot and either ringing or an
exponential decay toward the initial conditions.
Of all the parasitic elements present, only the two of major significance,
that is, L e and C„ will be included in the linearized models. The stray
capacity, which is distributed throughout the circuit, may be lumped
into a single element. But where should it be placed? If it is inserted
376 switching [Chap. 12
from the grid to cathode or directly across the magnetizing inductance,
nothing constrains the plate voltage; once the tube turns on, the full
supply voltage appears across L e and the plate immediately bottoms at
zero. This will not occur in the physical circuit. If the model permits
such a drop, the model is incorrect. To prevent the abrupt change in
ej, the stray capacity should be connected from the plate to cathode as
shown in Fig. 1211.
Without actually solving for the exact time response, we shall now
discuss the basic behavior of the circuit and the role of the various
parameters.
Region I. Initial Rise. The voltage swept through during the
initial portion of the grid rise, from cutoff to zero, is a very small per
centage of the total voltage change. Moreover, since the grid will not
load the plate circuit, the loop gain will be much higher than that meas
ured in the positive grid region. The rate of voltage change is very rapid,
and the time contribution to the overall pulse duration insignificant.
In the positive grid region, the nature of the response would be found
from the model of Fig. 1214a. Since
ei
e c = = — %inr c
n
the circuit poles would be given by the solution of the determinant
R + w. Anr °~w.
or
, , / 1 , n*rA , R + w 2 r, + nAr e _ „ n „ 9 „,
p* + (rc, + t;) p + — bljc. ° {12 " 23)
For proper switching, one pole must lie in the right half plane. With the
values given in Fig. 1211, the two poles are located at
Pl = 7.43 X 10 7 p 2 = 2.43 X 10 7
In a relatively short time the effect of the negative exponential will
have damped out. The time response is primarily due to the positive
pole, and we may approximate the platevoltage fall by
From the definition that the rise time is the time required to charge
from 10 to 90 per cent of the bounding voltage,
fx = — = 0.091 /usee
Pi
Sec. 126]
THE BLOCKING OSCILLATOR
377
The negative pole, along with the additional parasitic elements not
included in the model, will slow the rise by a factor of 1.5 or 2. Any
increase in L, or C, would also adversely affect the switching time.
I'M'r
Ebb nEcc
(c)
Fig. 1214. Models holding for regions I and II of the vacuumtube blocking oscillator
of Fig. 1211. (a) Incremental model holding within the active region; (6) model
denning the rise in the saturation region; (c) model used to solve for ft,min and E c , m „.
This portion of the response is finally bounded when the grid rise and
plate fall drive the tube into saturation. Reflecting the grid circuit
voltages into the transformer primary results in
En + nE cc = e b + e L + ne c
where ej, is the drop across the leakage inductance. Solving for
e h = e e = Eb\
we obtain
En =
Ebb + nE cc — 6l
n + 1
(1224)
Equation (1224) cannot be solved until e L is evaluated. But this is too
difficult to do. The particular voltage at which the tube saturates is
not crucial to the circuit operation. Because L e is small, the voltage
developed across it will not be large even at the boundary. Arbitrarily
assuming e L = 10 volts, the circuit used as an example yields
Ebi =
200  40  10
= 50 volts
Region II. Final Current Buildup. After the tube saturates, the
circuit model reduces to the one shown in Fig. 12146. Because R L
is very much larger than the 285ohm plate saturation resistance, the
3,000ohm external load may be neglected. The two poles in the left
378 SWITCHING
hand plane are now located by the solution of
r p . + n 2 r c
or at
p . + (_L_ + V^\
V + \r P .C + L e )
pi = 9.51 X 10 7
P +
P2
r P .L.C,
[Chap. 12
(1225)
1.75 X 10 7
Charging continues from the boundary value of E e i = En toward the
steadystate conditions, which can be found from the model of Fig.
1214c. The pole located closest to the origin is associated with the
longest time response, and consequently the duration of this charging
interval is approximately
iPil
= 0.228 Msec
From Fig. 1214c the peak voltages are given by
Eh,aLi * = r , + n*r c ^ Ebb "*" ^
Ee.nuti = ; — y— (Etb + nE^.)
r ps + n*r c
(1226a)
(12266)
Solving these equations in the circuit used as an example, we find that
the plate falls to 20 volts while the grid rises to +70 volts. In an actual
circuit r c would decrease markedly when the grid voltage greatly exceeds
e t . This lowers E c , aiI and raises E htaia .
C
HI
Fig. 1215. Model holding over the timing region.
Region III. Pulse Timing. This, the most important region, is
denned by the very simple model of Fig. 1215. Note that because
L m » L e , C, is much less significant and may even be omitted. The
magnetizing current begins increasing with the time constant
n'r e
toward the steadystate value
/.. =
Ew>
Sec. 126] the blocking oscillator 379
Meanwhile, the plate voltage rises from jE t , mill toward Eu, and the grid
decays back toward E cc . The two charging equations are
e„ = E»,  (B»  E h , mia )e'i* (1227o)
e„ = Ecc  (E C c  Ec.^Je'i* (12276)
Eventually the decay at the grid and the rise at the plate will permit
the tube to reenter the active region. Solving Eqs. (1227o) and (1227&),
the time elapsed until ei, = Em = E c % is
h = t In „ „ (1228)
Ebb ■+■ Jbcc
The value of E ai = E c % may be found by substituting the U back into
Eqs. (1227) or by eliminating the exponential term between these two
equations. The simplest expression for the boundary value becomes
Ec2 = E i2 = Etb + *f" (1229)
We might observe that the 53 volts at which the circuit leaves the satura
tion region is slightly higher than the 50 volts at which it entered. The
duration of the pulse is
h = 20 X 10* In (1 + 5 %2o) = 20 X 10~« In 1.227 = 4.08 j*sec
At the end of the pulse the magnetizing current has reached a peak of
/m(f 8 ) = — (1  e"") = 130 ma
Tp.
If the transformer cannot tolerate such a current without saturating,
the pulse would terminate proportionately earlier.
In practical blocking oscillators, the high grid current which charges C
increases the rate of the grid voltage decay. Switching will occur
somewhat sooner than given by Eq. (1229). This same charging is
also used in the astable circuit to backbias the tube during the interpulse
period.
Region IV. Termination of the Pulse. Once the circuit again becomes
active, regeneration drives the tube toward and even beyond cutoff.
As long as the tube remains active, the poles defining the response will be
given by Eq. (1223). Below cutoff the time response is defined by the
parallel Rl, L m> and C, circuit. Depending on the degree of damping,
the output may be either an exponential decay or a damped sinusoid.
The final decay is from a peak determined by the energy previously
stored in L„. The external damping Rl not only limits the backswing
to a safe value, but also prevents the ringing that may cause a false
retriggering of the circuit.
380
SWITCHING
[Chap. 12
The magnetizing current of 130 ma flows through Rl after the tube is
cut off. This results in a peak voltage at the plate of
#w = Eu, + I m (h)R L = 590 volts
At the grid the maximum possible backswing is
£ .„i. = 20  39% = 215 volts
The above calculation neglected the stray capacity, which will, of
course, slow the switching and reduce the amplitude of the backswing.
Fig. 1216. Plate and grid wavi
Kg. 1211.
for the monostable blocking oscillator of
127. Some Concluding Remarks. Even a small receiving tube,
limited to 2 or 3 watts plate dissipation and 0.5 watt grid dissipation,
is able to supply 50 or 100 watts of peak power over the narrow pulse
interval. To ensure that the average power will not exceed the ratings,
the blocking oscillator is normally operated with a duty cycle of 1 to
5 per cent (time on as compared with the total period). For pulses hav
Sec. 127]
THE BLOCKING OSCILLATOR
381
ing a larger duty cycle, multivibrators would be used in preference to the
blocking oscillator.
On the other hand, only a small amount of power is dissipated by
the saturated transistor during the pulse. If large currents are switched
in the collector circuit, then the peak collector dissipation is reached
within the active region — during the turnon and turnoff times. Larger
duty cycles are possible without damaging the transistor. Usually the
resistance inserted to limit the voltage backswing imposes much more
severe restrictions on the duty cycle (long recovery time) than do the
power considerations.
Of the many other modes of blockingoscillator operation possible,
one of the most common is where the transformer primary is designed
to resonate with the stray capacity. The circuit operates as a tightly
coupled tuned plate oscillator such as discussed in Chap. 14. This
circuit is not really a relaxation oscillator, but it can still be analyzed
by the methods discussed in this chapter, with, however, the activeregion
model defining the major portion of the response. The waveshapes
appearing at the plate and grid would be portions of a sinusoid, although
somewhat distorted by the changing grid resistance.
PROBLEMS
121. In Example 121 the stray capacity is increased until the circuit is on the
verge of becoming oscillatory during recovery.
(a) Calculate the minimum value of C s which will resonate with L„.
(6) When C, is ten times the minimum value, sketch the response in the recovery
region. Pay particular attention to any retriggering of the blocking oscillator.
122. Figure 1217 shows an emittercollectorcoupled blocking oscillator where
the period is independent of any powersupply variations. Except for Ri and Ri, all
components are as given in Example 121.
Fio. 1217
382
SWITCHING
tCHAP. 12
(a) Prove or disprove the above statement.
(6) iJi is 500 ohms, Ri is adjustable, and C 2 is very large. Plot the peak value of
emitter current and the pulse duration as functions of ijj. At what value will the
circuit stop operating?
123. Figure 1218 shows a collectorbasecoupled blocking oscillator with the
external load connected to a tertiary winding.
(a) What is the minimum value of n for guaranteed operation? Indicate the
polarity of the transformer connections.
(6) Assume that the transformer turns ratio is two times the bounding value. In
addition, the ratio L m /L, = 25:1. Specify the transformer parameters for a 5Msec
pulse duration.
(c) Sketch and label the load current.
(d) Compare the advantages and disadvantages of this configuration with the cir
cuit of Fig. 121.
124. Draw the circuit for an emitterbasecoupled blocking oscillator. Using the
parameters of Fig. 1218 with leakage and magnetizing inductance (referred to the
emitter winding) of L m — 10 mh, L, = 0.5 mh, and n = 3 (base winding), calculate
the pulse duration. Sketch and label the load current and the collector voltage
waveshapes.
9+20 v
C10/if
rii=40D
/350
A 1,000
lv
Fig. 1218
126. The blocking oscillator of Example 121 is modified to the circuit of Fig. 125.
Repeat the waveshape calculations if the bottoming value is 2 volts. The Zener diode
chosen to limit the collector voltage conducts at 40 volts; in series with it, we insert a
recovery resistance to limit the peak at the collector to 75 volts. How would the
circuit respond if the Zener resistance is 10 ohms? ohms?
126. Show three methods of triggering the blocking oscillator of Example 121.
Use both current and voltage input pulses. Specify the pulse height, polarity, and
order of magnitude of the source impedance. Does any one method chosen have
overwhelming advantages? Explain.
127. The transformer of Example 121 is changed to one that saturates at 50 ma.
Assume that L m drops from 2.5 mh to 250 jm after saturation. Plot the new wave
shape, paying particular attention to operation after saturation. Are all the assump
tions made in the text still valid? What happens after the transistor turns off?
128. Figure 1219 shows a blocking oscillator designed so that the pulse width
can be controlled by a constant current injected into a tertiary winding. This cur
rent establishes a bias flux and thus influences the saturation point. Saturation
THE BLOCKING OSCILLATOR
383
occurs at a field strength of 50iV X 10~* ampturns, where N is the number of turns in
the collector winding. Specify the parameters (L m , C, and B) that will permit the
adjustment of the pulse duration from 5 to 50 psec. Neglect the effect of L, on the
pulse width.
<?+20v
0.98
ri'i_40Q ci
1219
129. (a) The circuit of Fig. 1218 is made astable by reversing the base bias bat
tery. In addition, L„ = 5 mh and n = 0.2. Calculate the value of C that will set
the period at 6 msec. Neglect L, in your calculations. Show the recovery portion
of the collector waveshape and the capacitor current.
(6) By how much would the charging current reduce the pulse duration? Make
all reasonable approximations in your calculations.
1210. Repeat the calculations for the vacuumtube oscillator discussed in the
text when C, = 0. Sketch the waveshapes and compare the times with those pre
viously found. Plot the path of operation of these two cases on the voltampere
characteristic and compare the location of significant points.
1211. Can eh in Eq. (1225) be evaluated from the known conditions of the prob
lem? If it can be, do so. If it cannot be, explain why not.
1212. (a) Justify the approximations made in finding h and t t in the text in
Sec. 126.
(6) If these simplifications were not made, would it be possible to calculate the
exact response? Explain.
1213. (a) Find the value of Bl, in terms of the other circuit parameters, that will
just prevent ringing in the recovery region (Fig. 1211).
(6) Evaluate this resistance for the circuit discussed in the text. Calculate the
backswing produced.
1214. Sketch and label the platecurrent, gridcurrent, and cathodecurrent wave
shapes for the blocking oscillator of Sec. 126. Calculate the power dissipated in
the tube over the pulse interval.
1215. Plot the peak grid voltage; the minimum plate voltage; the peak grid, plate,
and cathode currents; and the pulse duration as a function of the transformer turns
ratio. Let 0.2 < n < 5 and use the example given in the text as your circuit. The
initial rise may be neglected in your calculations. Can any conclusions be drawn as
to the "best" transformer?
1216. An astable vacuumtube blocking oscillator is shown in Fig. 1220. Calcu
late the pulse duration and the period. Sketch and label 2 full cycles of the grid
voltage waveshape.
384
100 K
SWITCHING
?+200v
[Chap. 12
3.3 K
£ m 25 mh
L e 500pb
CSOjyif
Fig. 1220
BIBLIOGRAPHY
Benjamin, R.: Blocking Oscillators, /. IEE (London), pt. IIIA, vol. 93, pp. 11591175,
1946.
The Blocking Oscillator, Wireless World, vol. 63, pp. 285289, June, 1957.
Linvill, J. G., and R. H. Mattson : Junction Transistor Blocking Oscillator, Proc. IRE,
vol. 43, no. 11, pp. 16321639, 1955.
Narud, J. A., and M. R. Aaron: Analysis and Design of a Transistor Blocking Oscil
lator Including Inherent Nonlinearities, Bell System Tech. J., vol. 38, no. 3,
pp. 785852, 1959.
PART 4
MEMORY
CHAPTER 13
MAGNETIC AND DIELECTRIC DEVICES AS MEMORY
AND SWITCHING ELEMENTS
In this chapter we shall be making use of just those properties of
magnetic and dielectric materials that were so annoying in linear circuits.
One of the more important is hysteresis, i.e., the dependency of the
present state on the past excitation, the material's memory. The other
characteristic, saturation, which allows several regions of operation,
permits the use of these solidstate devices in controlled gates and other
circuits previously restricted to diodes, tubes, and transistors. Not only
are special materials used, but these are also treated to exaggerate the
hysteresis and saturation regions before being fabricated into cores or
capacitors. The behavior of magnetic materials has been studied much
more extensively, and consequently there is a much wider range of
properties available than can be found in the dielectric mediums. For
this reason we shall concentrate mainly on those circuits making use of
ferromagnetic devices. However, as this situation will most probably be
rectified in the near future, the last sections of this chapter will consider
certain ferroelectric devices and circuits.
131. Hysteresis — Characteristics of Memory. We must sometimes
store the information contained in a sequence of pulses which are gener
ated by a multivibrator or blocking oscillator and transmitted through
various gates. In digital computers, rapid storage and readout are
essential for the solution of complex problems. Besides storing data and
answers, the program controlling the sequence of the solution is fed into
the memory banks. As each step is completed, the computer switches
to the next memory location for further instructions.
Storage consists in switching a bistable device from its low to its high
state, arbitrarily designated by and 1. Thus a time sequence of pulses
may contain almost any desired coded information. If a pulse is present,
it would be indicated by 1, its absence noted by 0. For example, one
particular fivedigit word might be 10011. Each digit contains one bit
of information, and each would have to be registered in a separate mem
ory location. Upon the application of the appropriate readout pulses,
the excited memory units would switch from 1 to 0, with the previously
387
388
MEMORY
[Chap. 13
stored information appearing as output pulses. Those in the state
remain undisturbed.
We shall now consider the particular characteristics required in the
individual memory unit.
1. It should have at least two definitive, widely separated levels.
Once excited from the lower to the higher, the memory element should
maintain the new state after the initial storage pulse has disappeared.
This precludes use of all nonlinear devices having continuous character
istics. We cannot identify the state of the diode (curve a in Fig. 131)
by looking at the zero excitation point.
,
1
(a) (b) (c)
Fig. 131. Three typical characteristics of bistate elements.
2. Both the process of storage and readout should require a finite
amount of energy. If the device switches levels with an absolute mini
mum input, what is to prevent it from switching erratically in the presence
of noise? The two states of curve b in Fig. 131 are interchangeable at
zero excitation; a device having this characteristic would be unreliable
for use as a memory element.
3. At any instant the state of the device must represent the past
history of the system as well as the present excitation.
4. In general, storage and readout should require different types of.
excitation. This is necessary because we do not want the memory to
shift from 1 to when a storage pulse is applied or to store a readout
pulse falsely by switching from to 1. Such differentiation might be as
simple as using positive and negative pulses for storage and readout,
respectively.
It follows directly from the above statements that hysteresis is essen
tial in any device used for information storage (curve c in Fig. 131).
It can be a natural property of the device, as in the ferromagnetic
cores or in various ferroelectric dielectrics, or it can be simulated by the
use of diodes and energystorage elements, as discussed in Sec. 29.
Bistable multivibrators, constructed either with tubes or transistors or
with the twoterminal negativeresistance devices of Chap. 11, also exhibit
Sec. 132]
MAGNETIC AND DIELECTRIC DEVICES
389
hysteresis, and as we know, the state of these circuits is controlled by
input triggers.
As an example, consider the multivibrator of Fig. 132. Assuming
that it is initially in the state (T 2 off and Ti on), the voltage at the
grid of Ti is positive, limited primarily by the resistive loading of the
conducting grid. R x , R z , and the positive grid resistance form a resistive
summing network. Ti cannot be turned off and T 2 on until the external
input ei is negative enough to bring the grid voltage to below zero, i.e.,
to bring the operating point to point A in Fig. 132. Regeneration then
causes the jump to point B. Even if ei is made more negative, it will
have no further influence over e 2 . When the input trigger is removed, the
output finally stabilizes at point 1. The righthand portion of the
En
*2
D
,
B
1
C
>
(b)
Fig. 132. Bistable multivibrator and inputoutput transfer characteristic illustrating
hysteresis.
hysteresis curve is traced upon the application of a positive readout
pulse, which resets the multivibrator to state 0.
Among the disadvantages of multivibrators for use as memory ele
ments is their large size. Even when using transistors, the sheer bulk
of the units needed to store 10,000 or 20,000 bits of information would
force us to turn to some other storage device. Furthermore, since each
multivibrator always has one element conducting, a large bank of them
requires an excessive amount of power. Any failure in the power supply
would automatically erase the stored information. For the above rea
sons, the use of multivibrators is restricted to systems requiring only a
small memory capacity and to subsidiary registers in the larger computers.
132. Ferromagnetic Properties. In order to explain, even cursorily,
the salient features of the BH curve of Fig. 133, we shall start with a
microscopic region and increase our range until it encompasses the whole
core. There exist, within the material, small domains where all the
atomic magnetic moments are aligned in some preferred direction. At
390
MEMORY
[Chap. 13
room temperature iron has six and nickel has eight easy directions of
magnetization which are related to the crystal structure. Separating
the domains are walls composed of atoms whose moments are skew to the
principal direction. Since, from a macroscopic viewpoint, the dis
tribution of magnetic moments is random, the material is normally
unmagnetized.
An external field increases the area of those domains magnetized in its
direction and causes the contraction of those that have other magnetic
moments. As the excitation is increased, portions of the walls become
j Slope Amax
Fig. 133. Portion of the magnetization curve, showing some of the important con
stants of the material.
unstable, jumping to the direction of forced orientation. The finite field
required to produce this change results in the hysteresis shown in Fig.
133. When all the atomic moments are essentially in alignment, we say
that the material is saturated; it then supports a magnetic flux density
B,.
The presence of the external field is necessary to maintain the align
ment of the magnetic domains. After the excitation is removed, they
relax to the nearest easy direction of magnetization. Instead of the
original random orientation, there now exists a component of flux in the
direction of the applied field, and this is called the remanence B r . To
completely demagnetize the material, we must apply a reverse field that
will overcome the internal coercive force H c .
Sec. 132]
MAGNETIC AND DIELECTRIC DEVICES
391
One important factor used to relate the field and flux is the magnetic
permeability,
, = & (13D
which is not a constant, but increases from a low value near the origin
to a maximum near the peak of the magnetization curve, decreasing
as the core finally saturates. Even though B„ B r , H c , and cr (the volume
conductivity) are also needed for a complete description, **»„ serves as a
convenient measure of the magnetic quality. Those materials having a
high permeability almost always have a narrow hysteresis loop. For
example, iron, which has a maximum permeability of 5,000, requires a
Fig. 134. Typical hysteresis curves and equivalent static piecewiselinear represen
tation (479 Mo Permalloy).
magnetizing force of 3.2 oersteds to develop the saturation flux density
of 16,000 gauss. A highpermeability alloy, such as Permalloy (79 per
cent nickel, 17 per cent iron, and 4 per cent molybdenum), may have a
maximum m of 100,000 or higher. This material would saturate at a
flux density of close to 8,000 gauss, when the field strength is approxi
mately 0.1 oersted. It has a coercive force of 0.05 oersted and a reten
tivity of 6,000 gauss. Note that the hysteresis loop of this second
sample (Fig. 134) begins to approach a rectangle, so much so that it
becomes valid to approximate the curve in the piecewiselinear manner
shown.
The above discussion was based on the response either to direct current
or to relatively low frequencies where the flux in the core has time to
build up to the maximum value permissible with the applied excitation.
If the dynamic hysteresis loop were to be plotted by using a veryhigh
frequency excitation signal, then the area of loop, which represents the
392
memory [Chap. 13
This effect is noticeable even at
kB
~H C
H c
H
losses, would be greatly increased.
400 cycles (Fig. 134).
In pulse applications we are interested, not in the steady state, but in
the much more complicated transient BH response. With any suddenly
applied magnetization force H a , we should expect an immediate flux
density B a . It would not be present. The H field must produce the
alignment of the magnetic domains before the flux given by the BH
curve can be sustained. In addition, the H field propagates from
the surface into the core with a finite velocity. Initially, only the
surface layer contributes, and therefore the effective y. is an extremely
small percentage of the final value. It will, of course, increase as the flux
builds up toward steady state.
Both the coercive force and the retentive field are structural proper
ties, depending on such factors as grain orientation and microscopic
imperfections, i.e., voids and inclusions. If, in working the magnetic
alloy, the grain size is controlled and
aligned to favor one easy direction of mag
netization, (!„„. can be greatly increased.
One method used to produce almost square
loop materials (Fig. 135) is to cool the
hotrolled sheet slowly in a hydrogen
atmosphere with the magnetic field applied
in the direction of rolling. The high per
meability is measured only in the direction
of grain orientation. To obtain the full
advantage, the thin tapes produced are
wound into toroidal cores. Deltamax (50
per cent nickel and 50 per cent iron),
which has a maximum permeability of 70 000, has a squareness factor
(B T /B.) of 0.98.
Energy must be supplied to the excitation coil to overcome the losses
involved in establishing and aligning the magnetic domains. The input
per unit volume required to switch between two points on the hysteresis
loop is given by
w = (£*' H dB (132)
The terminal characteristics of the coil depend on the amount of stored,
and hence recoverable, energy. In switching from point 1 to 2 in Fig.
133, the net energy supplied is proportional to the area between branch
I and the B axis. Once the excitation is removed, the operating point
traverses segment II from point 2 to 3. Only the energy represented by
the small area between branch II and the B axis is recoverable. The
area of the hysteresis loop lying in the first quadrant represents the
Fig. 135. Square hysteresis loop
such as found in a tapewound
toroidal core of Deltamax.
MAGNETIC AND DIELECTRIC DEVICES
393
Sec. 133]
dissipated power, and consequently the resistive component of the ter
minal characteristic.
In squareloop materials (Fig. 135), all the supplied energy is dissi
pated and none can be recovered. We thus interpret the drivingpoint
characteristic of the core as that of a nonlinear resistance. Since this
case is of great interest, it will be considered in some detail below.
Ferrites. Some of the disadvantages of the thinstrip cores have been
overcome by the use of various molded ceramic semiconductors. These
materials, such as NiOFeuO, (nickel ferrite) or MgOFe 2 3 (magnesium
ferrite), have a much lower electric conductivity and consequently lower
losses. The nature of the material is such that the domain switching
time would be much less than that of the tape cores. Even though the
maximum permeability is small (m„« = 1,000), it is almost independent
of frequency. The easily fabricated squareloop ferrite cores, of almost
any desired configuration, are widely used in highspeed memory arrays
and switching circuits.
133. Terminal Response of Cores. Before we can make use of cores
in specific circuits, we must know their terminal response to various
excitations. We have seen that
this is a function of both the core
geometry and the shape of the
hysteresis loop. Because the gen
eral problem is quite complex, our
discussion will concentrate on a
squareloop core of relatively simple
geometry. The core of Fig. 136
is constructed by winding a thin
tape of grainoriented magnetic
material on a bobbin. By making
the following assumptions the sub
sequent analysis will be greatly
simplified.
1. At every point the radius of curvature is large compared with the
thickness d. In any small region the core may be treated as an infinitely
long plane sheet and the geometry reduces to two dimensions.
2. The width h is much greater than the thickness d. Fringe effects
may be ignored. Consequently the excitation can be assumed to be a
current sheet flowing on the outer and inner surfaces of the core.
3. The hysteresis loop is perfectly rectangular. It follows that the
core, which is initially magnetized to — B„ switches to +B, as the H
field propagates from the surface toward the center.
4. Because of the symmetry, the field will penetrate at equal rates
from both the inside and outside of the core.
Fig.
m
136. Tapewound core.
394
MEMORY
[Chap. 13
Once the magnetic field is established on the surface of the core, it
creates the domain wall, which
propagates toward the center,
dividing the region having a flux
density of +B, from that at — B..
Throughout the complete material,
the flux will have one of these two
values. Only across the domain
wall will there be any change of
flux with respect to either time or
space. Figure 137 illustrates the
conditions in the core before the
field penetrates to the center and
completes the switch in the satura
tion direction.
Response to a Known Voltage
Excitation. The terminal voltage
Fig. 137. Cross section of the core illus
trating the flux penetration and the
direction of the current flow.
response of the excitation coil is given by
«,(<) = N^ = 2Nh£ / Bdx
dt dt Jo
But from Fig. 137, we see that
fd/2 /g] \
/ Bdx = B,x, — B, f k — x, J
dx,
and consequently
v{t) = ANhB,
dt
(133)
(134)
(135)
Solving Eq. (135) yields the spacial variation of the domain wall in
terms of the known voltage excitation.
x.(t) =
4NhB,
(136)
Saturation occurs when the domain wall reaches the center of the core.
Setting x, = d/2, we obtain
j^' v(r) dr = 2NB.M = 2NB.A
(137)
where A = crosssection area, m 2
B, = saturation flux density, webers/m s
Equation (137) states that the time required for the complete reversal
of the core state depends on the volttime area of the applied voltage
signal. Regardless of waveshape, equal area signals will result in equal
Sec. 133] magnetic and dielectric devices 395
switching times. Or expressing this result in another way, under ideal
conditions a volttime area greater than that given by Eq. (137) cannot
be sustained across the excitation coil.
In the special case where v(t) is a constant V, Eq. (136) reduces to
x,(<) = WW.
and we can see that the domain wall propagates at a constant velocity
from the surface to the center. The time required for complete satura
tion is
t. = ?*£* (138)
In order to calculate the current flow, we turn to Maxwell's equations.
Because of the simple geometry, they reduce to
^ =  a E (139)
dx
dE = _dB
dx dt
(1310)
where E = electric field intensity
a = volume conductivity of the core material
From Eqs. (1310), (134), and (135),
The presence of E(t) results in an eddycurrent flow in the excited portion
of the core. This current may be considered as establishing a retarding
field which limits the time rate of flux reversal.
B is constant everywhere except across the domain wall. Hence the
spatial variation of the electric field must be zero everywhere except at
x = x,. In the center of the core, where the magnetic field has not as yet
penetrated, E(t) will be zero (x > x,). For x < x„ the electric field
will be given by Eq. (1311). Since E is constant with respect to x, H
must decrease linearly from the surface value H a (t) to the coercive force
H c , &tx = x,. It follows that Eq. (139) may be rewritten as
MLl^ = M ^='J (1312)
x, dt 2Nh
where H equals Ni/l, ampturns/m, in the mks system.
Substituting B, dx./dt and x„ as given by Eqs. (135) and (136), into
Eq. (1312), we obtain the solution for the terminal current due to the
396
MBMOEY
[Chap. 13
+ /.
(1313)
known voltage excitation.
** (<)= 8™.j/ WdT
The first term in Eq. (1313) is the total eddycurrent flow in the core;
the second term corresponds to the current needed to overcome the
internal coercive force. We might note that the higher the conductivity
of the ferromagnetic material, the
greater the core loss. If the ap
plied voltage is a constant, the
current will increase linearly with
time as shown in Fig. 138.
m
.
fk
Ic
SiKV 2 t+I c
ia{t) =
ll
8NWB.
VH + I c
Fig. 138. Terminal current response of a
squareloop core to a constantvoltage
excitation.
After the core saturates, the cur
rent will continue to increase, but
at a rate controlled by the induct
ance of the driving coil in air, its
winding resistance, and the internal resistance of the voltage source.
We might further note that the constantcurrent term in Eq. (1313),
due to the coercive force, may be treated as a separate bias generator in
parallel with a postulated ideal core (zero width hysteresis loop). This
leads to the conclusion that the response of the core may be represented
schematically by the models of Fig. 139.
^tffflifflT
(a)
o
h N
ww
R(v,t)
O
(b)
Fia. 139. Proposed equivalent model representations for the squareloop core.
Response to a Known Current Excitation. As our starting point, we
shall solve Eq. (1312) for the location of the domain wall in terms of the
applied magnetic field.
*® = aSI
[H.(t)  H c ] dr
Substituting x,(t) back into Eq. (1312) yields
2Nh[H a (t)  H c ] VK
v(t)
V<r/ '[ff o (r)H c ]dr
(1314)
(1315)
Sec. 133]
MAGNETIC AND DIELECTRIC DEVICES
397
When the input is a constant current I a , the terminal voltage and the
saturation time are given by
v(t)
4
t. =
i 4hWB.(I.  /«)
alt
<rlB,d 2
42V(/„  L)
(1316)
(1317)
According to Eq. (1316), the terminal voltage must be infinite at
t = 0. This impossible solution is a result of the assumed ideal core
characteristics. In practice, a finite
time must elapse before the domain
wall is established. Furthermore,
the stray capacity present also
limits the time rate of voltage
buildup. The actual response will
look like the dashed rather than the
solid line of Fig. 1310. After the
core saturates at t = t„ the two
terminals should appear to be a
short circuit and the voltage should
drop to zero. However, the large
series impedance of the current
source together with the small air
inductance of the driving coil actually produces the fast exponential
decay shown.
From Eq. (1317), we can see that an external excitation of /„ would
not allow the core to saturate until an infinite time had elapsed. The
larger the excess of current over the internal bias value, the faster the
core will switch between the two saturation values. Here also the
coercive force may be treated as an external bias generator as shown in
Fig. 139.
Response of an Externally Loaded Core. The voltage appearing across
a load R connected to a secondary winding of N t turns is
v B (t) = Nivit)
where v(t) is the terminal voltage developed across the excitation wind
ing. An additional retarding field of
t, t
Fig. 1310. Voltage response to a con
stantcurrent excitation of the core.
H s (t) 
Rl
(1318)
is produced by the current flow in R. Ht(t) acts in the same direction
as the internal coercive field, and it is simply added to H, in all the
integral and differential equations previously derived.
398 memoey [Chap. 13
In the case where the applied voltage is a known function of time, the
input current given by Eq. (1313) would include, on the righthand side,
the term
N 2 v(t)
ir =
R
On the other hand, when a known current waveshape is used for
excitation, the timevarying ff 2 must be included inside the integral
sign. The voltage response would have to be recalculated from Max
well's equations. We would find that the smaller the effective load
resistance, the larger the additional retarding field and the longer the
time required to switch the core's state.
From an examination of the final response to both constantvoltage
and constantcurrent excitation [Eqs. (1313) and (1316)], we see that
the nonsaturated core might be approximated by a nonlinear resistance
of the form
R < = m < 13  19 >
in parallel with the current generator /„ (Fig. 139). K has one value
for constantcurrent excitation and yet another value when a step of
voltage is applied.
The total resistance seen at the driving terminal is the parallel com
bination of the timeandvoltagedependent core resistance R c and the
reflected load resistance.
Ri.
= R (jjj \\R e at t = 0, R c = oo, R^ = UjJ R
The now finite resistance limits the initial value of the voltage. The
parallel combination is always smaller than the input resistance of the
unloaded core, and consequently the voltage produced by a constant
current drive will be less than that found in Eq. (1316). In fact, if
RiN/Ni) 2 ^C R over the complete time and voltage range of interest,
then the input resistance would remain essentially constant and a rec
tangular voltage pulse would be the result of a constantcurrent excita
tion. Since the switching time is controlled by the volttime area [Eq.
(137)], the external loading will have the adverse effect of increasing t,.
To a good approximation, the ferromagnetic core can be considered
to be a reasonably large resistance when it is unsaturated and a very
small resistance (practically a short circuit) when saturated. It thus
acts as a bivalued circuit element with, however, its switching action
dependent on its own past history. If multiple windings are available,
then the current flow in one can be used to control the terminal response
at the others. For example, by alternately reflecting a high and a low
Sec. 133]
MAGNETIC AND DIELECTBIC DEVICES
399
resistance into the transmission path, the magneticcircuit element can
replace the diode bridge of Sec. 36 for use as a controlled gate (Fig.
1316).
A Note on Units. The parameters of the ferromagnetic sample are
usually expressed in the electromagnetic units (emu) of oersteds (H) and
gauss (B). However, all equations used and derived are given in mks
units. The important conversion factors are:
Symbol
Emu
Mks
Magnetizing force
Flux
Flux density
Permeability
H
B
Oersteds X 10V4*
Maxwells X lO"" 8
Gauss X 10*
Gauss/oersted X 4x X Vf
Ampturns/m
Webers
Webers/m*
Henrys /m
Example 131. In this example we shall compare the current response of two cores
to an excitation of 20 volts. The first core is toroidally wound on a }£in.diameter
form of 20 turns of 1milthick 0.25in.wide Permalloy tape. The second core is a
molded ceramic ferrite of the same size. Both cores have a 200turn excitation
winding.
gauss
Br,
gauss
oersteds
« *
jiohmcm
8,000
4,500
7,200
3,400
0.07
0.18
55
Ferrite
10 s
*p is the volume resistivity, the reciprocal of the volume conductivity <r.
Solution,, (o) The first step in the solution is the conversion of the mixed units
given in the statement of the problem into the mks system, using the conversion fac
tors given above. The parameters of Permalloy are H c = 5.56 ampturns/m,
B, = 0.8 weber, and p = 55 X 10 8 ohmm. Expressing the dimensions in meters
and substituting into the appropriate form of Eq. (1313) results in
t„(«) = 1.45 X WH + 1.16 X 10* amp
From Eq. (138) the saturation time of the Permalloy core is
/, = 51.8 jisec
At the end of the switching interval the coil current has increased to
t.(«.) = 760 + 11.6 ma
where the first term in the above expression is the eddycurrent flow and the second
is due to the coercive force. We might note that the low conductivity of the Permal
loy sample results in relatively large eddy currents. This means that additional
power is dissipated during the switching interval above and beyond that necessary
to overcome the coercive force.
(6) We shall assume that the results derived for a tapewound core apply equally
well to the sintered ferrite core. They actually do not, but the answers obtained are
400 memory [Chap. 13
on the correot order of magnitude and may be used for comparison. The solution of
Eq. (1313), for this sample, is
j.(0  14.1i + 4.1 X 10"» amp
The ferrite core switches state in only 29 jisec. Hence the peak current flow becomes
».«.)  0.41 + 41 ma
Because of the much higher resistivity of the ferrite sample, the eddy current of 410
/ia is completely negligible. However, because of the large coercive force, the required
driving current is still appreciable.
134. Magnetic Counters. The basic magnetic counter of Fig. 1311
consists of two components: the core itself where the count is registered
by the change in the magnetic state and an energystorage element
which automatically resets the core
after the required count has been
t entered. One of the more common
v applications of this circuit is as a
1 binary, where one output pulse is
I produced for every two inputs. In
r
Fiq. 1311. Basic magneticcore counter. this respect the magnetic counter
behaves similarly to the vacuumtube
or transistor bistable circuits of Chap. 9. The circuit of Fig. 1311 does
offer one major advantage in that no power is required to sustain the
excited state.
So that we may make use of the results of Sec. 133, we shall assume
that the core is constructed of perfect squareloop material. In its
unexcited condition, it is at —B,; we shall call this state 0. A positive
input pulse drives it toward +B, (state 1). The differential equation
describing the response to the applied excitation is
iJ
v iu = Ri + ~ I idt + v, (1320)
where v, is the voltage drop across the coil.
In Sec. 133 we have seen that the core appears to be a very high
resistance during the interval when it is switching from to 1 or from
1 to 0. Since R is the small series combination of the inputsource
impedance and the winding resistance of the coil, almost the full excita
tion voltage will be developed across the driving coil. Equation (1320)
reduces to » in = v„ Hence the initial response is due to the known
voltage excitation of the core and the current is given by Eq. (1313).
Until such time as the core saturates, the very small output may be
approximated by
v„(t) = £ idt
Sec. 134]
MAGNETIC AND DIELECTKIC DEVICES
401
In the special case of a constantamplitude pulse, this reduces to
Vo(t)
^ (KVH* + Ij)
(1321)
where K is evaluated with the aid of Eq. (1313).
The volttime area of each input pulse V At is adjusted so that the
core state will not reverse completely. The first one primes the core,
leaving it in the partially magnetized condition corresponding to point a
in Fig. 1312o. After the pulse terminates, the capacitor discharges
through the high coil resistance. But since only a small amount of
charge was stored when the pulse was present, the demagnetization path
follows the dashed line in Fig. 1312a from a to b.
B
,
!
B s
'
■ — ■ * ■
b
a
H c
H c
B s
H
V
.
^A(^
I
t
i
I
V
1
t
I
i
\rc
It=RC
^A*>
<
) t x t 2 t 3 t
(a)
(b)
Fig. 1312. Response of the counter of Fig. 1311. (a) Hysteresis curve showing the
partial demagnetization during the discharge of the first pulse; (b) excitation pulses
and the resultant output.
The primed core also looks like a high resistance to the first portion
of the second pulse. Consequently, in the interval from to h (Fig.
13126), the output will be given by Eq. (1321). Once the core saturates
at +B„ its terminal response is that of a short circuit. The capacitor
continues charging toward V with the small time constant RC. It is
only here that a large amount of energy may be stored in C in a short
time interval. It follows that R must be kept as small as possible.
Otherwise the single output pulse, which represents the count of two
(Fig. 13126), may never even reach V.
At the end of the second pulse, the core is left in state 1. But as the
capacitor starts discharging from its peak voltage V, the reverse current
flowing in the driving coil switches the core back to state 0. Any energy
remaining after the core resaturates at —B, is dissipated in the series
resistance R. This portion of the cycle is identified by the double
segmented decay waveshape shown in Fig. 13126. The interval from
402
MEMORY
[Chap. 13
h to t» represents the flux reversal, and the remaining portion shows the
final discharge of C. To ensure the automatic reset after every second
input, the energy stored in C must be greater than that dissipated in the
core and the series resistance. Neglecting the small power loss in the
series resistance, we obtain the necessary inequality
YiCV* > 2B.EM
4B,HcAl
or
C >
(1322)
If Eq. (1322) is not satisfied, then the discharge of C will not completely
reset the core to state 0. The next applied pulse starts the switching
process from a partially excited condition. It may even drive the core
directly to state 1, leading to a false count at the output.
We conclude that the core in the magnetic counter acts as a controlled
series gate, with its bivalued state dependent on its own past history.
Reset
9
8
5
4
3
—12
. .1
Set
H
(a) W
Fig. 1313. (a) An arbitrary base counter using a pulse shaper; (6) path of operation
followed in a scaleof10 counter.
When unsaturated, it acts as a high series resistance which prevents the
charging of the storage capacitor. After the core finally saturates, its
resistance drops to zero, thus closing the charging path.
Counting to an Arbitrary Base. If the volttime area of the input pulse
is carefully controlled, then this same circuit may be used to count down
by an arbitrary factor. Assuming that the minor hysteresis loops are also
reasonably square, each input pulse shifts the core state in discrete jumps
along the hysteresis curve, as shown in Fig. 13136. The final excitation
produces saturation and resetting. For example, a scaleof10 counter
will result when each input pulse has a volttime area that is slightly more
than onetenth of the total needed to reverse the saturation direction.
The extra area of each pulse causes the core to saturate near the beginning
of the tenth input and consequently permits the storage of the reset
energy in C during the final count.
Provided that the input signal is larger than the minimum necessary to
switch the core state, the volttime area of the signal developed across a
secondary winding will depend solely on the turns ratio and the core
Sec. 134]
MAGNETIC AND DIELECTRIC DEVICES
403
properties [Eq. (1317)]. Only in the unsaturated condition does the
ferromagnetic device act as a transformer; once it saturates, the driving
voltage will be developed across the series resistance instead of across the
low input impedance. This means that a second core may be used to
shape the variable input pulses to a constant area. The charge stored in
C, in Fig. 1313a serves to reset the shaping core after each pulse. The
diode determines the polarity of those applied to the counting core; it
prevents the reset pulse of the input core from registering as a negative
count in the storage core.
Cascading Counters. In order to achieve higher counts than are
possible with a single core, the output of one stage may be coupled to a
second, and so on. Each core represents the appropriate binary or
decimal place. Because of the difficulty in reading the counts correspond
ing to the unsaturated states of the core (numbers between 1 and 9),
most cascaded counters use a binary rather than a decimal base.
Fig. 1314. Threestage cascaded counter.
When several counters are directly coupled, as shown in Fig. 1314,
the single input must be able to supply the energy needed to reset all cores
simultaneously. Consider the operation when the three cores are in
state 1 (just below saturation) and the count is 1 X 2° + 1 X 2 X + 1 X 2 2 ,
reading from left to right. The eighth pulse saturates core 1, and as d
charges, core 2 saturates, and so on, until finally C 3 is also fully charged.
Thus we see that the volttime area of the input pulse must be able to
change the state of all cores and that it must also supply the energy
that will reset all three cores.
If more than two or three stages are cascaded, they may impose a
severe load on the signal source. Consequently, an active regenerative
circuit is usually included within the transmission path to decouple and to
reshape the transmitted pulse. These circuits are discussed in Sec. 135.
Shift Registers. Suppose that we wish to store the digital word 10011.
Since it is equivalent to
10011 = 1 X 2* + X 2 s + X 2 2 + 1 X 2 l + 1 X 2°
this word can be converted into a train of nineteen pulses, which would
then be injected into five cascaded binary counters. But as this requires
an additional step, it increases the possibility that an error may be
404
MEMORY
[Chap. 13
introduced in the count. Moreover, if the word could be registered
directly, less time would have to be allotted for storage.
When the five digits are simultaneously available, each can be steered
directly to the appropriate core. This form of data processing, where
only one pulse width is needed to register the information, is called
parallel readin.
On the other hand, the word 10011 may be presented in series as a
time sequence (the serial input is pulse, pulse, blank, blank, pulse).
Output
Shift pulse
Input
pulses
Reset
(shift)
pulses
Binary
Delay
Binary
Delay
(b)
,.
Input
,
! i '• !
i i — i i ! i — i ' i i — i
t
Shift
i
i
i
1 i
1 i
i I
i i
t
M
Fig. 1315. (a) Magneticcore shift register; (6) block diagram of any shift register;
(c) time relationship between the input pulses (10011) and the shift pulses.
Separating each digit there is sufficient space to insert the pulses that
control the switching of the shift register of Fig. 1315a. The first
digit, i.e., the one farthest to the right in the word to be stored, is inserted
in the normal way into core a, changing its state from to 1. The
decoupling diode prevents the resulting negative output from switching
the second core in the chain.
Before the next digit appears, we apply a separate shift pulse to all
cores (Fig. 1315c). Those that are in the primed state will be switched
back to 0, while those already in the zero state will be unaffected. We
see that the counter is set by the input and reset by the shift pulse.
The positive voltage, which will now be developed across the output
Sec. 134] magnetic and dielectric devices 405
winding of the switched core, charges the storage element C. Its dis
charge, through R and the primary of core 6, switches the second core
to state 1. In order to prevent the shift pulse from interfering with the
transferred digit, there must be a time delay in the forward transmission
path. In the shift register of Fig. 1315a, this delay is obtained by
inserting a resistance in series with the driving coil of the core.
The next digit of the word, input digit 1, again changes the state of
core a without influencing any other core. A shift pulse will now set the
excited cores a and b back to zero. As both capacitors discharge, the
two digits are transferred to the right, changing the states of cores b and c
from to 1.
The third digit of the input word is zero. After its application the
state of the cores is Oil — , where the dashes represent the unknown states
due to any previous excitation. A shift pulse transfers the core states to
001 1. Continuing the process of first switching the state of core a and
then transferring the stored information to the right, the full word is
finally registered.
Any other binary element that can be triggered on and off at two
separate inputs, or with two different polarity pulses, can replace the
magnetic cores in the register (Fig. 13156). Voltage or currentcon
trolled negativeresistance devices are ideally suited for use as memory
elements in shift registers. If bistable multivibrators are used, a simple
RC network can be used to delay the transfer until after the shift pulse
terminates.
For readout, five shift pulses will sequentially feed the stored digits
to the output terminal. This process is identical with the normal shifting
which transfers the input forward from core to core. In fact, as a new
word is entered (starting with a shift rather than a digit pulse), the old
word is automatically shifted out on the right.
One obvious disadvantage is that the stored word is erased as it is
fed out for examination or use. However, if the shift register is formed
into a closed loop, then the information fed out on the right can be auto
matically restored in the lefthand core. After the appropriate number
of shift pulses are applied, the register again contains the original word.
A convenient means of closing the selfstorage path is by way of a
controlled gate such as shown in Fig. 1316. This circuit makes use of the
bivalued resistance properties of the core to interrupt or close the feed
back path. During storage the gate is kept open (the gating core is in its
active region) and the register functions normally. When reading out
information temporarily stored, the gate is also left open. But if we
desire more permanence, then the gate must be closed (coresaturated)
during the intervals that the output pulses are produced. It is implicitly
assumed that the amplitude of the transferred pulse will not change the
406
MEMOBT
[Chap. 13
state of the magnetic gate and that the reflected control signal will not
register as a storage pulse.
Ring Counter. A closedloop shift register can also be used as an
arbitrary base counter. One and only one core (or binary) is set to
state 1. Then as each succeeding input pulse is applied to the shift
windings, the single stored digit is transferred sequentially around the
loop. For example, of the 10 cores necessary in a decimal counter,
initially only the zero core is excited. The first pulse transfers the single
digit to core one, and so on. The tenth input switches the ninth core
from to 1 and resets the zero core to 1. This same voltage pulse may
also be coupled to a second ring counter, which is used to indicate the next
highest decimal place.
Control signal
Input
Sp Output
Fig. 1316. Closedloop shift register — a switched core used as a controlled gate.
It follows that the particular count registered is indicated by the
number of the single excited core. If the volttime area of the transfer
pulse is carefully controlled, then the core can be switched from —B.
to the verge of +B,. In these circumstances the difference in the termi
nal resistance seen across an auxiliary winding readily identifies the
location of the digit. In fact, if the current flow through the readout
winding is always less than J c (so that it cannot influence the magnetiza
tion), then the individual core will act as a simple controlled gate,
inserting a high resistance during the count and a low one at all other
times. The gating process can, of course, be reversed by transferring
a rather than a 1 around the loop.
135. Core transistor Counters and Registers. The number of stages
which can be directly coupled in a counter or shift register is limited by
the power capabilities of the pulse source. In the counter, the input
must be able to switch all cores simultaneously. In the register, the
shift pulse not only resets all cores but must also supply the energy
that transfers the stored information to the right. And since the switch
Sec. 135] magnetic and dielectric devices 407
ing time depends on the excess of magnetizing current over the coercive
force [Eq. (1317)1, the problem is further complicated when the counting
speed becomes critical. For example, in order to switch a typical core
in a 250kc register, the ljisec shift pulse applied must have a peak power
of almost 1 watt. The 330ma pulse current will develop an average
of 3 volts — and a peak which may be many times as large — across the
shift coil. Under these conditions we can expect a single transistor
amplifier to switch, at the most, three or four stages.
One obvious means of improving the response is to use an amplifier
to decouple the individual cores. An even better way is by inserting a
regenerative circuit between two cores ; the output of the first core can be
made to trigger the generation of a fastrising highpower pulse for
propagation in the forward direction.
JL *
Input
Output
Bias
Bias
Fro. 1317. Cascaded counter using a transistor blocking oscillator for regeneration
and a core for storage. The input amplifier stage is also shown.
Figure 1317 shows a counter which is a happy combination of a core
and a transistor. This circuit employs a single core both as a memory
(winding Ni) and as a transformer for the blocking oscillator. Besides
the economy in using one rather than two cores, the sharing improves the
response because the oscillator pulse also, resets the counter. In the
normal state the core is at and the transistor is cut off by the external
bias. The first pulse brings the core to the verge of saturation, and the
second one saturates it and charges C x . With the transformer connec
tions shown, these input pulses act in a direction that keeps the transistor
off.
After the termination of the second input, the voltage across N\ reverses
as Ci starts discharging. This pulse, coupled to N 2 , brings the transistor
into its active region. And as the regenerative circuit goes through one
complete cycle, the extra energy supplied by the collector current rapidly
returns the core to state 0.
In the counter of Fig. 1317, the functions of regeneration and storage
408 memory [Chap. 13
are separately handled by the transistor and the core. The active
element is operative only during the reset interval. At all other times,
the passive core controls the circuit behavior. The capacitor only
triggers the regenerative circuit; it does not have to supply the energy
for reswitching. Hence the restrictions on C are much less severe, and
this circuit will recover much faster than those depending only on the
stored charge for reset.
From which point in the circuit shall we take the output? If the
collector is chosen, all the transformed input pulses will also be coupled
to the next stage. However, as these are of the opposite polarity to the
pulse produced by the blocking oscillator, a series diode will prevent
them from registering falsely at the next core. This diode may be
eliminated if the output is taken at the junction of the collector resistor
and the transformer winding, as shown in Fig. 1317. Here the cutoff
transistor effectively replaces the decoupling diode.
As an alternative, the output may be taken from a fourth winding
on the core. Since the switching during recovery is between the two
saturation levels, this pulse would have the stabilized volttime area
which is especially necessary when counting to a base other than 2.
By properly choosing the turns ratio, the output volttime area can be
adjusted to any desired percentage of the reset pulse. This avoids the
use of a separate core for shaping the counting pulses that are applied
to the next stage.
If, instead, the fourth winding is used as an input for the shift pulses,
then the train of cores and transistors will function as a shift register.
When the core is in the zero state, the low impedance offered to the shift
input will not permit the development of a pulse large enough to bring
the transistor out of cutoff. After the information pulse primes the core,
the opposite acting shift pulse can trigger the regenerative circuit. Dur
ing reset, the previously stored information is transferred one position
to the right.
136. Magnetic Memory Arrays. Figure 1318 illustrates the arrange
ment of a compact largecapacity memory unit. We shall assume that
each of the 25 small ferrite cores exhibits the ideal squareloop properties
of Fig. 1319. Before information is stored by changing its state, the
individual core is saturated at —B,. Three insulated leads thread the
core: one passes through each row, a second through each column, and a
third threads the complete array.
Suppose that we wish to switch the state of the single core 34, located
at x = 3 and y = 4. Positive current pulses of amplitude 7„ are simul
taneously applied to row 3 and column 4. The total field in the core
located at the intersection of the excited row and column (due to 27 a )
must cause switching. However, as all other excited cores should remain
Sec. 136]
MAGNETIC AND DIELECTBIC DEVICES
409
Fig. 1318. A 5 X 5 magneticcore coincident memory,
in their original state, the pulse amplitude must satisfy
y 2 Ic </.</„ (1323)
Only core 34 is excited past H c (to point 2H„ in Fig. 1319). All the
other cores in the row and column have a field strength of only H a ;
after the termination of the pulse they relax back to the original state.
Thus the injection of the two pulses carrying the address of the core
switches its state and stores the indi
vidual digit. These pulses may be
directed properly with the aid of
auxiliary steering cores or diode gates.
In order to read out the stored
digit, the memory unit must be inter
rogated. To do so, negative current
pulses ( — la) are injected into the
appropriate row and column. If the
core situated at the intersection is in
an excited state, it will switch back
to 0. The voltage developed during
the transition appears at the termi
nals of the readout winding which
threads all cores (Fig. 1318). Suppose that the particular core interro
gated is in the zero state. In this case the low impedance presented
#,
B s
H a
2H a
H c
B.
H
Fig. 1319. Hysteresis loop showing
excitation of switched and non
switched cores of the memory array
of Fig. 1318.
410
MEMORY
[Chap. 13
would permit the development of only a very small output, which could
not possibly be mistaken for a digit. Since only a single core is ques
tioned at a time, there will not be any ambiguity as to the location of the
output information.
We should note that the process of readout is destructive ; it erases the
stored information. External circuitry has been devised which auto
matically resets the excited core. The presence of the output digit
triggers the generation of a positive current pulse immediately after the
termination of the negative interrogation signals. Nondestructive stor
age and readout depending on special core geometries have also been
devised, but a discussion of these is beyond the scope of this text.
Switching time in the array of Fig. 1318 is on the order of 1 /*sec.
In order to reduce the access time, smaller cores must be used. But
doing so increases the problem of wiring the large arrays needed. Instead
of using individual cores, many highspeed memory units depend on
switching small regions of ferrite or magnetic film. The conductors are
first printed on an insulated sheet. At each "intersection" a small
amount of ferrite or metallic film is deposited. Holes punched through
the "core" region provide for the
insertion of the readout winding.
With the small amount of magnetic
materials used, switching times of
20 to 50 m/isec have been obtained.
A further advantage is that this
type of memory lends itself to the
economical production techniques
of automation.
137. Core transistor Multivi
brator. In Sec. 133 we have seen
that the saturation time of a
squareloop core, excited by a con
stant voltage, is very clearly de
fined. The abrupt change in the
drivingpoint impedance, which
marks the end of the flux buildup,
permits the core to be used as the
timing element in a monostable or an astable circuit.
Figure 1320 shows a singlecore multivibrator employing resistive
coupling to complete the regenerative loop. In this circuit, the winding
direction is such that when Ti is saturated, the current through 2Vi
drives the core from — B, to +£,. Current through the second winding
Nz, due to saturation of T 2 , resets the core state to — B,. As in all
astable multivibrators, the transistors furnish the greaterthanunity
rvws
iMMJ
i — nfflS^
nmp — i
iV 2 •
Fig. 1320. Coretransistor multivibrator
used as a dc to dc or a dc to ac con
verter.
Sec. 137]
MAGNETIC AND DIELECTRIC DEVICES
411
loop gain needed to switch between the two quasistable states. They
become active only during the small switching interval corresponding to
the saturation of the core. At all other times the individual transistor
is either saturated or cut off.
We shall now detail the operation over one complete cycle, assuming,
as the starting point, that Ti is off, just turning on, and T* is on, just
turning off. This is equivalent to saying that the core is on the verge of
saturating at — JS,. As long as the flux is still changing, the full supply
voltage will appear across the high drivingpoint impedance of N% and
the collector voltage of T* will be approximately zero. Hence no current
can flow into the base of T t ; it is cut off.
En ]_
VA
±
if I VW1 °
fa)
2E bb
<b)
Fig. 1321. Models holding during the change of flux density from — B, to B. in the
core of Fig. 1320. (o) Saturated transistor T\\ (6) backbiasing of cutoff transistor TV
Once the core saturates, there is a marked decrease in the coil's driving
point impedance. Under these circumstances an appreciable voltage
is developed across the collectortoemitter saturation resistance of the
series transistor 5P». Current can now flow into the base of the off
transistor, and as T\ turns on, the current through the winding iVi starts
the flux reversing. En is connected across this winding, as shown in the
model of Fig. 1321a. The transformer action reverses the voltage
across JV 2 . The resultant large increase in the base current of Ti rapidly
drives this transistor into saturation. At the same time, the drop in the
collector voltage of T\, coupled through the biasing and speedup capaci
tor Ca, will turn T 2 off (Fig. 13216).
The time that Ti remains saturated is controlled by the flux buildup
in the core. If the small demagnetizing base current is neglected, then
the core switching time, as given by Eq. (138), is
h =
2NiAB.
Ehj,
412
MEMORY
[Chap. 13
After h sec the core saturates at +B, and the circuit switches to its other
quasistable state.
The charge stored in the speedup capacitor aids in maintaining the
base of the off transistor below zero, thus stabilizing the circuit's opera
tion. From Fig. 132 la we can see that as long as Ti is saturated,
Ci is in series with the transformer winding JV 2 and Em,. When JVi = iV 2 ,
d will charge toward 2E bb . Meanwhile C 2 is discharging through R 2
as shown in Fig. 13216. The conditions which must be satisfied in
order to sustain the base of the off transistor below zero for the full half
cycle are
4r 2 > h 4n > h
where t 2 = R 2 C 2 and n = R x d.
Suppose that C 2 is too small; will its discharge before the end of the
half period cause the premature switching of this circuit? Since the base
Fig. 1322. Transistor collector and base waveshapes — circuit of Fig. 1320.
of the off transistor T 2 is returned to the collector of the saturated one,
T h only a very small base current can possibly flow into TV Its collector
current, flowing in winding iV 2 , cannot institute regenerative switching.
It only opposes the flux buildup due to the much larger current in
Ni. We thus conclude that the core still controls the timing.
If the two timing coils are identical, this circuit will generate the
square wave shown in Fig. 1322. The frequency of oscillation will be
given by
'  k  wjb, (13  24)
when the transistor switching time is neglected. It is interesting to
note that the squarewave frequency is linearly dependent on the supply
voltage.
The output is usually obtained from a third winding; hence the trans
Sec. 137]
MAGNETIC AND DIELECTBIC DEVICES
413
Output
Fid. 1323. A transformercoupled
coretransistor multivibrator.
former action not only affords isolation but also permits the amplitude
to be scaled up or down.
Figure 1323 shows another coretimed multivibrator, one which
utilizes transformer rather than resistive coupling between the active
elements. In this circuit, the volt
ages developed across the auxiliary
windings Ni and Nt sustain the one
transistor in cutoff and the other in
saturation during the flux buildup
interval. Consider the conditions
when the saturated transistor Ti con
nects Ebb across winding N* With
the directions shown, winding iVi sup
plies the saturation base current of
T\. The voltage appearing across
Nt backbiases Ti. These voltages
are easily calculated once the turns ratios are known.
The half cycle terminates when the saturation of the core permits the
transistors to reenter their active regions. Any decrease in voltage
across the driving coil is coupled by the transformer into the base of T 2 ,
turning it on. Since the circuit is regenerative, switching continues until
Ti saturates and Ti cuts off. The flux then starts reversing, and the
cycle repeats.
These same circuits are widely used as dc to ac or as dc to dc con
verters. The primary peaktopeak voltage of 22?«, (measured from
collector to collector) is first multiplied by the primary to secondary
winding ratio n. If direct current is desired, the square wave, which is
present at the output, must be rectified. Thus a 10volt battery may be
used to develop a 100volt (or even higher) ac or dc output. Of course,
the primary source impedance is multiplied by n 2 as it is reflected into the
output circuit while the voltage is only increased by the factor n. Thus
the PR losses increase faster than the voltage. This limits the per
missible stepup ratio, especially when large amounts of power must be
supplied to the load. In any case, power transistors having a very low
saturation resistance should be chosen.
When this circuit is used as a dc to ac converter, the frequency of
operation is usually the critical factor. It will even determine the
transformer design, i.e., the turns ratio, crosssection area, and type of
magnetic material employed. Since the frequency is inversely pro
portional to the product of the transformer parameters, the greater the
turns ratio, the smaller the volume of iron needed. However, increasing
the number of turns also increases the winding resistance and the losses;
■therefore some compromise must be selected.
414 memory [Chap. 13
On the other hand, when operating as a dc to dc converter, the
higher the frequency of oscillation, the easier it is to filter any ac com
ponents appearing in the output. As a further advantage, both the
amount of transformer iron and turns needed decrease with increasing
frequency. But if the frequency of operation becomes excessively high,
the stray circuit capacity will adversely affect the waveshape. Since
the transistor will remain active over a larger portion of the cycle, the
effective source impedance also increases, resulting in poorer regulation
with respect to load changes.
For example, a typical converter used in an automobile will operate
from the 12 volt battery and will supply a maximum of 1 amp to the
load at a nominal 120 volts and 60 cps. In this application the total
primary resistance may be estimated at 0.1 ohm, 0.05 ohm due to the
saturation of the power transistor and the other 0.05 ohm contributed
by the winding and battery resistance. Since the transformer's turns
ratio is 10: 1, 10 ohms will be reflected into the output. At full current
flow, the output drops to 110 volts. The primary voltage falls by the
same percentage and causes a frequency shift down to 55 cps [Eq. (1324)].
While this converter might be used to operate many small appliances, the
frequency change under load would prevent its use for phonographs or
other small constantspeed motors.
138. Properties of Ferroelectric Materials. Dielectrics are those
substances where all charged particles are relatively tightly bound to the
atomic nucleus or the molecular
region. In the presence of an ex
ternal field, these charges cannot
move freely through the material
to the surface as do the relatively
loosely bound electrons in conduc
tors. Instead, there will be a slight
separation in the individual micro
scopic region; the positive charge
Fig. 1324. Individual dipole. moVeS in the direction of the E
field, and the negative charge
moves in the opposite direction. Hence the local effect on the field
pattern is that of ah elementary dipole, i.e., an associated positive and
negative charge (q) separated by some small distance d (Fig. 1324).
In certain dielectrics these dipoles exist even before the external field
is applied. But since they are randomly oriented, the material does not
exhibit a net field of its own. It follows that the drivingpoint character
istics of the dielectric may be determined from the motion and alignment
of the elementary dipoles under the influence of the applied field.
Figure 1324 shows an individual dipole from which the potential field
Sec. 138] magnetic and dielectric devices 415
at any point can be calculated. If the point is relatively far from the
charge, the potential is given by
gdcos^ yolts (13 . 25)
where to, the permittivity of free space, is equal to 8.85 X 10 12 farad/m
in the mks system. The dipole moment is defined as
p = qd
and p is directed from the negative to the positive charge. This vector
is called the polarization of the dipole. In any small volume, where
there are n dipoles, the total polarization is the vector sum of the indi
vidual dipole moments. When they are all aligned, P = np, and when
they are in random array, P = 0.
In order to express the terminal characteristics in simple terms, we
define the displacement vector
D = *>E + P , (1326)
The relationship between P and E is, in general, quite complex; for
example, the polarization in a single direction in a crystal may depend
on all three components of the applied field. However, for the ideal
dielectric, we can assume the linear relationship
P = to kE (1327)
which leads to
D = «<,€ r E
where e r = 1 + fc is called the relative permittivity (or dielectric constant)
of the medium. In the ferroelectric materials with which we shall be
concerned, fc 2> 1, and the electric displacement is almost exactly equal
to the polarization.
One class of dielectrics can be defined as those materials where the
polarization is produced solely by the external field. For these to be
useful in capacitors, the relationship between P and E should be the
linear one expressed in Eq. (1327). If it is not, then the dielectric
constant, and hence the capacity, will vary with the applied voltage.
Certain organic waxes exhibit a quasipermanent residual polarization
when they are solidified in the presence of an electric field. These are
called electrets, and they are the closest electrostatic analogs to the
permanent magnet currently existing.
In the third and, for our purposes, the most important class of dielec
trics, the polarization is a nonlinear function of E. Here the material
is characterized by spontaneous polarization, saturation, and hysteresis
(Fig. 1325). So as to recognize the duality with the magnetic materials,
416
MEMORY
[Chap. 13
these dielectrics are unfortunately called ferroelectric. They contain
no iron, nor is the duality complete.
Ferroelectricity was first observed in rochelle salts, NaK(C4H 4 6 )
4H 2 0, by J. Valasek, in 1921. The
usable temperature range of this
substance is only 40°C, and because
of its complicated crystal structure,
it develops ferroelectric properties
in one direction only. Barium
titanate, BaTi0 3 , is much more
useful since it is ferroelectric be
low its Curie point of 120°C. The
crystal has a simple perovskite
structure. Many other substances,
such as triglycine sulfate and guan
idine aluminum sulfate hexahy
drate, also exhibit ferroelectric
properties. Some of these other
Fig. 1325. Typical hysteresis loop of
ferroelectric materials.
materials have squarer hysteresis loops, higher Curie temperatures, or
better longterm storage properties than barium titanate.
The switching behavior of the ferroelectric crystal has been extensively
investigated with the aid of polarized light and optical examination. In
the unsaturated sample, small domains are found where the net dipole
moment is in one of the permissible directions. Studies with a thin plate
(a) (b)
Fig. 1326. Domain switching, (a) 180° domains showing the spike formation; (b)
90° domains showing the wedge formation. The shaded regions have been switched
by the external field.
cut from a single crystal of barium titanate indicate that adjacent
domains may have polarization directions either 180 or 90° apart. In
the 180° domains, the application of a field E > E c causes the formation
of spikes of new domains in the preferred direction (Fig. 1326a). These
then extend across the whole crystal. The 90° domains are reversed by
the formation of wedges skewed to the cathode wall (Fig. 13266), which
grow and spread sideways, eventually covering the entire region.
The number of both the 90 and 180° wedges formed, and hence the
Sec. 139]
MAGNETIC AND DIELECTRIC DEVICES
417
switching time, is a function of the excess of field strength over the
internal coercive force.
139. Ferroelectric Terminal Characteristics. The practical form of
the ferroelectric device is a thin wafer of barium titanate, or some other
ferroelectric material, cut from a single crystal. The two electrodes
must be carefully deposited on the flat surfaces because any air gap that
exists will decrease the effective dielectric constant and seriously degrade
the squareness of the observed hysteresis loop.
^
^
v d
D.
Q.
V c
Vc
V
E c
E c
~e"
D,
Q.
(b)
(a)
Fig. 1327. (a) Schematic representation of a ferroelectric device; (6) ideal square
loop characteristics. Both the microscopic (E,D) and macroscopic (Q„F.) coordi
nates are shown.
In a parallelplate capacitor of thickness d and area A,
D =
Q
and
E =
V a
(1328)
where V a is the terminal voltage. Furthermore, the charge flow is simply
«/:
idt
For a known current excitation, the time required to reverse the satura
tion direction of a perfect squareloop dielectric (Fig. 1327) is given by
Q«
= 2Q, = T' i dt = 2D.A
(1329)
Equation (1329) states that a fixed currenttime area (charge) is
required to switch the dielectric. Or, expressing the relationship of
Eq. (1329) in a more useful manner, the ferroelectric device acts as a
series charge regulator. It permits only the fixed charge packet Q m
to flow during the switching intervals. This equation should be com
pared with the equivalent relationship previously derived for the ferro
magnetic core [Eq. (137)]. Under the special condition where the
418 memoky [Chap. 13
applied current is constant (t = /«), the time required to switch from
— D, to +D., or vice versa, is
. _ 2D.A
During this interval the terminal voltage increases linearly with time,
starting from the coercive value V c .
After switching, the ideal ferroelectric device can no longer transfer
any charge. Consequently, it seems reasonable to approximate the
saturated response by an open circuit.
/constant
V— constant
t.
(a)
(b)
Fio. 1328. Terminal response of ideal ferroelectric device,
excitation; (6) constantvoltage excitation.
(o) Constantcurrent
When a voltage excitation is applied, the switching time is found
to be inversely proportional to the excess over the coercive force. For a
constant voltage V a , the relationship is
t. =
K
V e
(1330)
where if is a constant that depends on the geometry and properties of
the ferroelectric medium. Equation (1330) is exactly analogous to
Eq. (1317).
To a very good approximation the voltage and current response of
the ferroelectric device during switching will look like the current and
voltage response of the ferromagnetic core, respectively (Fig. 1328).
Where our concept of the core was a timevarying resistor decreasing
from infinity as the core approached saturation, our model of the square
loop ferroelectric device is a resistance that increases from zero with time
and current. It appears in series with a bucking voltage equal to the
internal coercive force V c . Sometimes the model of the ferroelectric
element is further simplified, by neglecting the energy loss during switch
ing, and it becomes a bivalued capacitor. In fact, when the termination
of the excitation leaves the ferroelectric element unsaturated, its incre
Sec. 1310] magnetic and dielectric devices 419
mental response is most like that of a large capacitor. Since the relative
dielectric constant drops from several hundred down to unity as the
material saturates, the capacity is reduced to that of the two deposited
electrodes in free space.
The widely used ferroelectric material barium titanate has a coercive
field strength of 1,500 volts/cm. Only by using extremely thin slabs
(0.033 to 0.167 mm) can the coercive voltage V c be kept between 5 and
25 volts. Over the transition interval the average resistance of the ferro
electric element ranges between 100 and 1,000 ohms. Because of the
large remnant polarization, 22 X 10 8 coulomb/cm 2 , a large charge
packet is transferred during switching, even from quite small elements.
1310. The Ferroelectric Counter. The principle of charge monitor
ing is employed in the counter of Fig. 1329. As the input squareloop
"in
?HQr
Output
Input
Fig. 1329. Ferroelectric counter.
ferroelectric element (FEi) is switched from negative to positive satura
tion by the leading portion of the input pulse, a charge packet of 2Q,
flows through Z> x into the storage element C. This increases the output
voltage by the definitive increment
AF„ =
2Q.
C
(1331)
The negative portion of the input pulse resets FEi to negative saturation
with £> 2 establishing the current path. During this part of the cycle
the backbiased diode Z>i prevents the discharge of the previously stored
count.
Note that the input pulses must be properly shaped. They must
have positive peaks greater than V c + V , as well as a duration greater
than the switching interval. The negative peak can be smaller, but
its magnitude must still exceed V c . Otherwise the ferroelectric device
will not switch and reswitch between the two saturation limits and will
not meter the full charge packet to the storage capacitor.
During the final count the change in the output voltage brings the
regenerative circuit into conduction. For example, the final change of
A V,, volts may be used to trigger a monostable multivibrator or a blocking
420
MEMORY
[Chap. 13
oscillator which propagates a single count in the forward direction into
the next stage. This same pulse will turn on the shunting transistor,
discharge C, and reset the counter.
The switching action of the counter can be greatly improved by the
simple modification shown in Fig. 1330, i.e., the replacement of C by a
second ferroelectric element. Each input pulse will be shaped into a
charge packet of 2Qi by FEi. This charge also flows into FE 2 and starts
it switching from — Q a2 toward + Q,2 If the second element is larger,
n charge packets (or input pulses) will be needed to saturate FE 2 .
Until such time as it does saturate, the ferroelectric element appears
7?E,
HQr
input
M
Kd 2
>R, ±,
Regenerative
stage
D 3
M
Output
CDFE 2
1
_J
Fig. 1330. Improved ferroelectric counter.
to be a low impedance and only a very limited voltage can be developed
across it.
If FE 2 becomes saturated during the nth input, then it may be replaced
by a very high impedance for the remaining portion of the pulse. The full
excitation will appear across the output storage element, and instead of
depending on identifying the small charge of AV„ volts, the large voltage
jump presents a definitive trigger to the regenerative stage. With the
modification of Fig. 1330 this counter can accurately register many more
counts than possible with the simpler circuit. Furthermore, since the
count is registered by a change in the state of the element and not by the
storage of charge, it will be less affected by leakage resistance. Counters
having rates as low as one per day and counting ratios as high as 30 : 1 or
40:1 have been successfully operated.
In order to reset the storage element, a negative pulse can be applied
through a diode from some point in the regenerative stage. However,
unless resistance is inserted in series with D 2 , the two input diodes would
act as a short circuit, preventing the resetting of FE 2 . Current will also
flow into FEi during the reset interval, but this only aids its recovery
toward negative saturation.
Example 132. Another interesting application of the pulse area shaping by the
ferroelectric and ferromagnetic devices is the tachometer shown in Fig. 1331. The
sharp input pulses obtained from the distributor in a sixcylinder 2cycle engine are
assumed shaped to switch and reset the chargemetering ferroelectric element. Their
Sec. 1311] magnetic and dielectric devices 421
amplitude and duration will, of course, be erratic. When switching from negative to
positive polarization, the input element delivers a charge packet of 20 X 10~* cou
lomb. We wish the highly damped dc voltmeter reading of 5 volts to correspond to
a speed of 5,000 rpm.
Dc
Input * C5fc R< (J voltmeter
Fig. 1331. A ferroelectric tachometer (Example 132).
Three input pulses are generated on each revolution. Hence the minimum spacing
between the individual inputs, which occurs at the maximum velocity, is
„, 60 sec .
= 4 msec
" 3 X 5,000
So that the storage capacitor can discharge completely between the adjacent pulses,
the output time constant must be much less than T m . Assuming that the charging
time is insignificant and that T > 4t, the average voltage read on the dc meter is
given by
V d .. = ± JJ *V e'/' dt = AV
T
where &V is the peak output voltage due to the metered charge packet. Note that
the output RC circuit provides a dc voltage inversely proportional to the spacing
between the pulses and directly proportional to the motor rpm.
At 5,000 rpm,
AF t = ^=r RC  20 voltmsec
Thus R = 1,000 ohms. Setting T = 500 yusec yields C = 0.5 rf. Since AV = 40 volts
and V c ■» 10 volts, for proper switching the peak positive input must be greater than
50 volts. The diode allows satisfactory resetting with negative peaks greater than
10 volts.
1311. Coincident Memory Arrays. The ferroelectric element is some
what more difficult to use in a memory matrix than the ferromagnetic
core. Since only two electrodes are available, these must be used not
only for storage and interrogation but also for readout. It is not pos
sible to place a third electrode at the memory position, as it was to thread
the complete core array with a third winding. Consequently various
ingenious circuits have been devised to obtain the output digit. %
Consider the memory plate of Fig. 1332, where the x and y direction
leads are printed at right angles on the opposite faces of the ferroelectric
plate. The intersections are the memory positions, and they may be
treated as individual ferroelectric elements. In order to obtain the
coincident switching, voltage inputs must be applied to the two leads.
422
MEMORY
[Chap. 13
To store a digit in a particular location, that memory element is switched
from negative to positive polarization by applying a positive pulse to the
j/i row and a negative one to the Xj column. Each pulse must have an
amplitude falling within the limits
YiV c <\v a \< v.
and a duration sufficient to cause switching.
V a 
4=U
TT
TT
ru
34C
11
1
Storage pulses
Readout pulses
*1 *2 x 3 *
Memory matrix
Fig. 1332. A ferroelectric memory matrix.
It is quite simple to reset the memory element from 1 to 0. If it
contains a stored digit, pulses of the polarity opposite to those used
for storage will sum at the intersection and reverse the polarization.
All other positions remain unaffected. But how may the digit be
registered after interrogation? The switching process results in the
transfer of a charge pocket which must somehow be monitored. In one
method, shown in Fig. 1333a, a resistor is connected in the ground return
of either the x or y pulse generator. The current flow during storage
will cause a negative voltage to appear across the resistor. During
reset, the opposite polarity signal is developed. The two output pulses
are easily distinguished, and if desired, a shunting diode may be incorpo
rated to eliminate the unwanted negative pulse.
Figure 13336 illustrates a somewhat superior method of producing
an output after interrogation. All the x leads (or y leads) are threaded
through a single readout core. The current flow during the transfer
of the metered charge packet induces a pulse into the output winding.
One p*olarity output is produced on storage, and the opposite on reset;
the diode shown selects the correct signal.
One major disadvantage of the ferroelectric memory matrix is that the
state of the particular element is affected by singlevoltage pulses of less
than the coercive force. In barium titanate it was discovered that
multiple queries in any one row caused the partial depolarization of all
MAGNETIC AND DIELECTRIC DEVICES
423
memory positions subjected to the halfvoltage pulse. After several
hundred interrogations the polarization may even be reduced to half
of the saturation value. It has been found, however, that crystals of
other materials, such as triglycine sulfate, appear to have more nearly
(a)
Fig. 1333. Two methods of monitoring the readout pulse.
ideal characteristics. As these sustain full polarization under repeated
questioning, they will no doubt be used in practical memory banks.
PROBLEMS
131. A squarecrosssection Permalloy core measuring J£ in. on a side has an effec
tive magnetic path length of 6 in. It is excited by a 10volt peaktopeak square
wave applied across a 20turn winding.
(o) What is the minimumduration square wave needed to drive the core between
its two saturation limits?
(6) Plot the spatial penetration of the field and the current buildup as a function
of time when the duration of the half period is twice as long as that found in part o.
In the course of the calculations assume a source impedance of 10 ohms and wind
ing resistance of 1 ohm where necessary to keep the solution within bounds.
132. Plot the spatial variation and current buildup for the core of Prob. 131 for
the following excitations. After the core saturates, the current is limited by the
100ohm series resistance; this resistance may be neglected during the interval that
the core remains unsaturated.
(a) A 50volt peaktopeak 30^secperiod sine wave.
(6) A 50volt peaktopeak 30jisecduration sawtooth.
(c) A 50volt peaktopeak 30/usecperiod triangular wave.
133. Consider the two models shown in Fig. 1334 which have been suggested to
approximate the core response during the switching interval. In order to see the
Fio. 1334
424 memory [Chap. 13
advantages and disadvantages of these representations, we shall compare the exact
and approximate response to a known voltage excitation.
(a) Calculate the model constants (R, k, and h) for the Permalloy core of Example
131. The current at t = t, of both models should agree with that given in the text
for constantvoltage excitation.
(b) Plot on the same graph the exact and the model current response to a voltage
ramp which increases at the rate of 5 volts/jisec. It drives the core from negative to
positive saturation.
Which model gives the best representation? Is there any modification of the model
which will result in a closer correlation? Explain.
134. Plot the incremental drivingpoint resistance as a function of time for the
two cores described in Example 131. Will this resistance change with excitation?
In your answer consider the response to a voltage ramp, which increases at a rate of
10 volts /Vsec.
135. Find an equivalent resistance for each of the cores of Example 131 that will
yield equal power dissipation over the switching interval. Treat the following two
cases and compare the results.
(a) Constantvoltage excitation (20 volts).
(6) Constantcurrent excitation (/ = 10/ e ).
136. Repeat Prob. 131 when the core is excited by a square wave of current hav
ing a peaktopeak value five times as large as the coercive force. In this case plot
the terminal voltage response and the location of the domain boundary with respect
to time.
137. Calculate and sketch the voltage response of the core of Prob. 131 to the
following current excitations. The core is initially at negative saturation, and each
signal reverses the flux direction.
(a) A current ramp which increases at the rate of 20 ma//isec.
(h) A single cycle of a sine wave having a peaktopeak amplitude of 50 ma and a
duration of 5 jusec.
138. (a) Solve for the expression of the voltage response to a unit step of current
when the core is loaded across a secondary winding N 2 with a resistor R.
(6) Plot the response of the Permalloy core of Example 131 to a 20ma unit step.
The external loading consists of 2,000 ohms across a 10turn winding.
(c) Repeat part 6 when the load resistance is decreased to 100 ohms.
139. In the counter of Fig. 1311 the ferrite core is wound with 20 turns; it has an
effective radius of 3.0 cm and a square cross section of 1.0 cm on a side (its parame
ters are those given in the text). C = 0.001 ^f, and R = 500 ohms.
(a) Specify the range of the volttime areas of the input pulses for which this
counter will count down by the ratio 4:1. What is the minimum spacing between
these pulses?
(b) Repeat for a counting ratio of 8:1.
(c) Sketch and label the output when the core is excited by a train of 20volt
5jisecwide pulses. These are widely spaced compared with the recovery time of
the core.
1310. A ferrite core having a radius of 0.5 in., d = 0.2 in., and h = 0.3 in. is used
as the memory element in the simple counter of Fig. 1311. One output pulse should
be produced for every three of the 25volt 10jusec input pulses. During the two
priming inputs, the output must remain below 5 volts.
(o) Specify the minimum value of C that will properly reset the core on the termi
nation of the count.
(b) Using twice the minimum value of C, specify the range of core turns for which
this counter will work. (The RC time constant is 1 usee.)
MAGNETIC AND DIELECTRIC DEVICES
425
(c) Sketch and label the output voltage when C is twice the minimum value and
when JV lies in the center of the acceptable range.
1311. Two identical ferrite cores having the dimensions given in Prob. 1310 are
used in the scaleof10 counter of Fig. 1335. Complete the design by specifying C.
Fig. 1335
needed to reset the input core. Calculate the minimum input volttime area and the
pulse spacing necessary for proper counting. Discuss the circuit modification neces
sary to ensure the automatic reset of the counting core.
1312. (a) Draw the circuit of a transistor multivibrator shift register, indicating
where and how to inject the input and the shift pulses. Show three stages. Indicate
the time delay between the stages.
(b) Repeat part a when a tunnel diode is used as the memory element. Pay
particular attention to the coupling network and to the method of injecting the input
and the shift pulses.
(c) Neon bulbs which fire at 100 volts, which have a conduction drop of 75 volts,
and which require 0.5 ma to sustain conduction might be used to indicate the count
in a core decade ring counter. Explain how they would be incorporated into the
circuit.
(d) Exercise your ingenuity to see if it is possible to ensure the reset of the core ring
counter, regardless of the stored digit, by the injection of a single pulse. If neces
sary, the basic circuit may be modified.
1313. Figure 1336 shows a magnetic core and gate incorporating automatic
reset. Each coil consists of 5 turns wound on the single ferrite core (radius of 3 cm
and with a cross section measuring 1 cm by 1
cm).
(o) Sketch the output voltage when 1msec
current pulses, having an amplitude equal to
0.4/ c , are simultaneously applied to two
inputs; to three inputs.
(6) If C » 0.001 A what is the minimum
pulse width that will ensure the automatic
reset of the core? The loss in R may be
neglected in the calculation.
(c) How might this circuit be converted
into an or gate?
1314. The transistorcore multivibrator
of Fig. 1320 is used as a dc to dc con
verter producing 150 volts from the 20volt source. It operates at approximately
1,000 cps, which simplifies the filtering at the output of the fullwave bridge rectifier.
Each of the 10turn primary windings has 0.1 ohm resistance. The saturation
resistance of the transistor is 0.2 ohm, and at the boundary between the active and
saturation regions, p may be taken as 5. Grainoriented silicon steel, havmg the
following parameters, is used for the square crosssection core: B, = 2 webers, B, =
12 ampturn/m, r = 40 X 10"» ohmm, and I = 0.3 m.
SJVk
Fio. 1336
426
MEMORY
[Chap. 13
0.05 mm
(a) Calculate the maximum value of R and the minimum value of C necessary to
sustain one transistor off and the other one on over the full half cycle.
(6) What crosssection area is needed for the core to operate at 1,000 cps?
(c) Calculate the total power dissipated in the converter when the external load
current is 150 ma. How much power is dissipated under noload conditions?
(d) Sketch and label the current and voltage waveshapes at the base and collector,
under loaded conditions.
1316. A siliconsteel core, having the characteristics given in Prob. 1314, is used
in the multivibrator of Fig. 1323. For this circuit the cross section measures 2 cm
on a side and the magnetic path is 20 cm long. The primary consists of four windings
of 5, 15, 10, and 5 turns, which are connected to the base and collector of the first
transistor and to the collector and base of the Second transistor, respectively. The
secondary is a single 50turn winding.
Sketch and label the collector and base current and the output voltage waveshapes'
when Ebb = 20 volts. Assume a basetoemitter resistance of 0.02 ohm for your
calculations.
1316. (a) Plot the potential as a function of 9 far from an individual dipole.
Assume that r is constant and equal to 50d.
(b) On the same graph as in part o plot the constant potential distance r as a func
tion of 9. The absolute value of the po
tential should be equal to the maximum
value found in part a.
1317. The incremental capacity of
0.2 mm the wedgeshaped ferroelectric element
shown in Fig. 1337 is a function of the
applied direct current. Calculate this
capacity variation, neglecting any fringe
effects. The dielectric used has t, = 2,000
Fio. 1337 and E c = 1,500 volts/cm, and its cross
section area is 0.25 cm. (Hint: First show
that the capacity of a largearea parallelplate capacitor is C = e e,A/d.) How
will the resonant frequency of a tank circuit, using this element, vary with the dc
voltage?
1318. (a) Prove that the charge packet flowing during the switching interval in a
ferroelectric element is equal to the charge transferred in a parallelplate capacitor of
the same dimensions as it charges from — V c to V c .
(6) The feiroelectric element FE t described in Prob. 1319 is charged from a
20volt source through a 5,000ohm resistor. After saturating, « r drops to unity.
Plot the current and voltage response. Calculate the approximate timevarying
resistor
B(t) **At+B
that will result in equal charge flow over the switching interval. Furthermore, the
terminal voltage at t = t, should be equal to V c 
1319. The counter of Fig. 1338 is excited by two input sources, each connected
to the single storage capacitor through a separate chargemetering element. FEi
consists of a 0.03mmthick slab having an effective surface area of 0.02 cm 2 , while
FE S is twice as thick and has only half the electrode area. The ferroelectric material
has a coercive field strength of 2,000 volts/cm and a remnant polarization of 8 X 10 _ «
coulomb /cm 8 . We may further assume that the constant in the switching equation
(1330) is K = 10 voltMsec.
MAGNETIC AND DIELECTRIC DEVICES
427
+20
v
20
*.
"20
EE? ~i I
Fig. 1338
Trigger
Sketch the output voltage, giving all significant values, if the Bhaped 20voltpeak
20/isecwide pulses are applied at the two inputs at
«i at t = 0, 10, 20, 30, . . . msec
e% at t  5, 15, 25, 35, . . . msec
1320. Figure 1339 shows a monostable multivibrator where the duration of the
quasistable state is controlled by the
switching time of a ferroelectric element.
In order to simplify the calculations it will
be represented by a 0.005/xf capacitor which
decreases by a factor of 1,000 after saturat
ing at ± 5 volts. Sketch and label the volt
age at the collector of Ti and at the collec
tor and base of T 2 after a pulse is injected
into the base of Ti. Does this type of
timing offer any advantage over that ob
tained from a simple capacitor?
1321. Show that the circuit of Fig.
1339 can be modified so that it can make
use of a ferromagnetic core to control the
timing. If the core of Prob. 1310 is employed, sketch and label the important
waveshapes.
BIBLIOGRAPHY
Bates, L. F.: "Modern Magnetism," 3d ed., Cambridge University Press, New York,
1951.
Chen, Kan, and A. J. Schiewe: A Single Transistor Magnetic Coupled Oscillator,
Trans. AIEE, pt. I, Communs. and Electronics, vol. 75, pp. 396399, September,
1956.
Chen, T. C, and A. Papoulis: Domain Theory in Core Switching, Proc. Symposium
on Role of Solid State Phenomena, Polytech. Inst. Brooklyn, April, 1957; also in
Proc. IRE, vol. 46, no. 5, pp. 839849, 1958.
Collins, H. W.: Magnetic Amplifier Control of Switching Transistors, Trans. AIEE,
pt. I, Communs. and Electronics, vol. 75, pp. 585589, November, 1956.
Dekker, A. J.: "Electrical Engineering Materials," PrenticeHall, Inc., Englewood
Cliffs, N.J., 1959.
Katz, H. W.: "Solid State Magnetic and Dielectric Devices," John Wiley & Sons,
Inc., New York, 1959.
Fig. 1339
428 memobt [Chap. 13
Little, C. A.: Dynamic Behavior of Domain Walls BaTiOs, Phys. Rev., vol. 98, no 4
pp. 978984, 1955.
Menyuk, N.: Magnetic Materials for Digital Computer Components, /. Appl. Phys.,
vol. 26, no. 6, pp. 692697, 1955.
Meyerhoff, A. J., and R. M. Tillman: A Highspeed Twowinding Transistormag
neticcore Oscillator, IRE Trans, on Circuit Theory, vol. CT4, no. 3, pp 228
236, 1957.
Rajchman, J. A.: A Myriabit Core Matrix Memory, Proc. IRE, vol 41 no 10 pp
14071421, 1953.
: A Survey of Magnetic and Other Solidstate Devices for the Manipulation of
Information, IRE Trans, on Circuit Theory, vol. CT4, no. 3, pp. 210225,
September, 1957.
Royer, G. H. : A Switching Transistor DC to AC Converter Having an Output Fre
quency Proportional to the DC Input Voltage, Trans. AIEE, pt. I, Communs.
and Electronics, vol. 74, pp. 322326, July, 1955.
Rozner, R., and P. Pengelly: Transistors and Cores in Counting Circuits, Electronic
Eng., vol. 31, pp. 272274, May, 1959.
Sands, E. A.: An Analysis of Magnetic Shift Register Operation, Proc. IRE, vol 41
no. 8, pp. 993999, 1953.
Storm, H. F.: "Magnetic Amplifiers," John Wiley & Sons, Inc., New York, 1955.
Von Hippel, A.: "Dielectric Materials and Applications," John Wiley & Sons Inc
New York, 1954.
Wolfe, R. M.: Counting Circuits Employing Ferroelectric Devices, IRE Trans, on
Circuit Theory, vol. CT4, no. 3, pp. 226228, 1957.
PART 5
OSCILLATIONS
CHAPTER 14
ALMOST SINUSOIDAL OSCILLATIONS—
THE LINEAR APPROXIMATION
The earlier chapters of the text showed how the generation of many
periodic waveshapes (rectangular pulses and linear sweeps) was predi
cated on driving the active elements of the circuit far into their saturation
and/or cutoff regions. The timing function depended upon the non
active regions of the system, with the active zone simply supplying the
energy necessary for switching between the two exponential timing
regions.
As we start discussing the generation of almost sinusoidal signals, it
should be pointed out that the nonlinearity of the system will now play
a subsidiary role. The basic timing is due to a frequencydetermining
network, with the amplifier simply setting the necessary conditions for
oscillation. To avoid distorting the output signal, the degree of non
linearity must be small, and consequently most of the subsequent dis
cussion can concern itself with the linear approximation of the sinusoidal
oscillator.
141. Basic Feedback Oscillators. Figure 141 presents the basic
configuration of a feedback oscillator as a block diagram. Even though
each of the three essentials shown
is not completely isolated, it is con
venient to think of them as sep
arate entities. It is especially
helpful to segregate the complete
nonlinearity so that it can be given
individual attention. The remain
ing elements in the circuit may then
be treated by the standard methods
of linear analysis.
The behavior of the circuit of Fig. 141 is defined by two equations,
one for the forward transmission and the other giving the feedback
voltage.
e» = («i + e f )AL e t = ffez
431
ei+ef
A
^ «2
Ue)
r~
e 3
1
_«
p(u>)
*
Fio. 141. Basicfeedbackoscillator con
figuration showing the ideal amplifier A,
the amplitude limiter Lie), and the fre
quencydetermining network /S(u).
432 oscillations [Chap. 14
From these equations, we find that the overall closedloop gain is
(? = £?= AL (14.!)
ei 1 — /SAL v '
For selfsustained oscillations, there must be an output without any
external excitation. This becomes possible only when the denominator
of Eq. (141) vanishes. It follows that the frequency and amplitude
of oscillation must satisfy
1  p(fi>)AL(e) = (142)
where 0AL is simply the openloop transmission (i.e., the switch in Fig.
141 is open). Equation (142) implies that the feedback must be
regenerative.
Limiting is often accomplished by driving one or more of the amplifier's
active elements into their saturation or cutoff regions for a portion
of the cycle. Alternatively, an amplitudecontrolled resistor or other
passive nonlinear element may be included as part of the amplifier or
in the frequencydetermining network. If a gross nonlinearity is per
mitted, the limiter will distort the signal and the output will be far from
sinusoidal.
For small signals no limiting will occur and L(e) will take on its maxi
mum value of unity. It follows that if Afi > 1 in the smallsignal region,
the amplitude will build up until the limiter stabilizes the system at an
output level that satisfies Eq. (142). Thus, in the active region, the
threshold loop transmission that permits selfsustained oscillation is
PA, = 1
This equation is called the Barkhausen criterion for oscillation.
Unity loop gain at a single frequency is a necessary but not a sufficient
condition for selfsustained oscillations. If the network characteristics
are such that the net phase shift is zero at several frequencies, then the
criterion for oscillation can be arrived' at only from an examination of the
complete amplitude and phase portrait of the system. Nyquist stated
that the polar plot of Ap for co < oi < a> must encircle the point
1 + jO for oscillation. However, for many of the simpler configurations,
the Barkhausen condition will be both necessary and sufficient. Below
we shall consider circuits that are known to oscillate and that can be
analyzed on the basis of their having unity loop gain.
To ensure our objective, the generation of a sinusoidal signal, the
frequency should be primarily determined by a network whose character
istics can be rigidly controlled. This network must contain at least two
energystorage elements so that the system response [Eq. (142)] will
have the necessary pair of complex conjugate roots which give the
Sec. 141]
ALMOST SINUSOIDAL OSCILLATIONS
433
natural frequency of oscillation. For simplicity, A may be assumed
to be a constant, with its frequency dependence lumped together with
that of 0.
Since A in Eq. (142) now represents the gain of an ideal amplifier,
it will be a pure number having a sign corresponding to the stages of
amplification (minus for an odd number and plus for an even number).
P represents the transmission characteristics of the frequencyselective
network and will therefore be a function of w. To satisfy Eq. (142)
the imaginary part of f)A must vanish. Since the complete frequency
variation is included in the /3 network, the imaginary part depends only
on j8. Consequently the frequency at which
Imp =
(143)
will be the approximate frequency of oscillation.
At the oscillation frequency, the remaining portion of Eq. (142),
that is, Re 03 A), must be identically equal to unity. The value of gain
that will just ensure the sinusoidal oscillations will be given by
or
Atfim)
A t
j8(«o)
(144)
X
T 1 I
This value of threshold gain is measured with the amplifier output loaded
by the input impedance of the net
work. The opencircuit gain must,
of course, be somewhat larger.
Equations (143) and (144), taken
together, state that the overall loop 
gain must be unity and the overall g*
phase shift must be zero at the fre 1
quency of oscillation.
Consider the circuit of Fig. 142,
where we shall assume that the
amplifier is adjusted so that it will
just sustain the oscillations. The frequencydetermining network is
characterized by
R
«z
Fig. 142. Example of a feedback oscil
lator.
m = % =
pCR + 1
pili R
n ^~ pG^ pCR + 1
3 + pCR +
1
pCR
Substituting ja for p results in
=
1
3 + i(cu/o) — Wo/«)
(145)
(146)
434
OSCILLATIONS
[Chap. 14
where a> = l/CR. The imaginary portion of the denominator of Eq.
(146) vanishes at
1
O)o =
CB
At this frequency £ = J^, and from Eq. (144) we find that the gain
necessary to sustain oscillation is A t = 3. To achieve this positive gain,
the amplifier must contain an even number of stages.
Because they determine the nature of the transient response, it is
interesting to examine the location of the roots of 1 — 0A, and hence
the poles of the complete feedback system as a function of A. The
roots of the circuit of Fig. 142 are found by substituting Eq. (145)
into 1  0A = 0. This leads to
P 2 +
3 A
CR
P +
(n)'°
(147)
Equation (147) has two roots which traverse the path shown in Fig. 143.
As A increases from to 1, they first
coalesce at —l/CR on the negative
real axis and then separate along the
semicircular paths shown, finally
reaching ±j/CR on the imaginary
axis for A = 3. This condition cor
responds to the threshold of oscilla
tion, with the location of the com
plex conjugate roots giving the
frequency.
When the system has two distinct
complex conjugate roots, i.e., for A
between the limits 1 < A < 5, the transient response will be given by
e(t) = Ke°" cos (out + <j>)
And the two roots are located at
Pl,2 = OC ± jbll
For small values of a, the natural frequency is
Fig. 143. Migration of the roots of
Eq. (147) as a function of A. The
arrows indicate increasing gain.
where a>o = y/a 2 + ui 2 .
almost o>o
If A is less than 3, then the real part of the root will be negative
and any oscillatory response will damp out. The degree of damping
depends on the distance from the poles to the imaginary axis and hence
on the value of A.
Once the poles move into the right half plane, the real part forces an
increasing exponential buildup. But the voltage is bounded by the
Sec. 142] almost sinusoidal oscillations 435
limiter; the output eventually stabilizes at a peak amplitude that makes
the average gain over the cycle equal to 3. At the peaks of the sine
wave, the limiter reduces the loop gain, driving the system poles from the
right into the lefthalf plane. The amplitude decays, the circuit reenters
the active region, and the roots move back into the right half plane.
If the poles are initially close to the imaginary axis, then the buildup
will be quite slow and small changes in the root location will have but
little influence over any single cycle of the sine wave. When they are
located far into the right half plane, the exponential buildup makes itself
felt during each cycle; the waveform will include a greater degree of
distortion. Even with a small gain margin over the threshold value,
the location of the roots will be such that the oscillator will generate a
signal of slightly different frequency than that calculated on the basis of
the borderline behavior.
The oscillator will be selfstarting only if initially the poles lie in the
right half plane. It follows that the necessary gain must be somewhat
in excess at A,. Any infinitesimal disturbance will cause an amplitude
buildup until the limiter stabilizes the output. The threshold gain
sustains existing oscillations, but does not provide a margin for buildup.
Excess gain will also stabilize the circuit in that any slight reduction in A
will not stop the oscillations when A > A t but will cause them to damp
out when the circuit operates marginally.
If the amplifier gain ever becomes large enough to drive the roots onto
the positive real axis, that is, A > 5, the system response will no longer
be oscillatory. The exponential buildup will lead to a relaxation
phenomenon similar to that found for the astable multivibrator.
142. Characteristics of Some BC and LR Frequency determining
Networks. The selfstarting oscillator drives itself into limiting and in
the process distorts the sinusoidal output signal. Harmonics, introduced
by the limiter, may be treated as additional signals injected within the
feedback loop. Each harmonic term will also be affected by the feedback
present ; it will be acted upon by the factor
where /3„ is the feedback factor evaluated at the particular harmonic under
investigation.
Since we wish to maintain an almost sinusoidal output, we should
ensure that the amplitude of the harmonics produced in the limiter will
be small compared with the fundamental. Each distortion term is further
modified by H n . With a properly designed network, the feedback will
change from positive to negative as the loop excitation frequency increases
from wo to 2wo. The magnitude of H„ will become less than unity at the
436 oscillations [Chap. 14
secondharmonic and even smaller at the higherharmonic frequencies.
Thus the nature of the existing feedback is changed and is used to reduce
the amount of distortion.
The more selective the network, the greater the degree of limiting
which can be tolerated without excessively distorting the final sinusoidal
output. Consequently .ffs serves as a measure of the network's quality
for oscillator application. For consistency it will always be evaluated
under threshold conditions, instead of using the slightly larger value
of the actual smallsignal gain. The best network, all other things being
equal, is the one having the lowest value of H 2 .
The output signal, which is of the form
e 3 = E l cos (coot + <t>i) + Y H„E n cos (nu4 +• <(,„) (149)
n = 2
is transmitted back to the input through the £ network. Each term is
also multiplied by /3„, resulting in an effective driving signal of
00
ei = faEi cos (wot + 00 + Y p n H„E„ cos (nuot + <*>„)
re2
In order to have a true basis of comparison as to the relative amplitudes
of the distortion components at the input and at the output, it is advisable
to use the normalized input harmonic factor
K. = ^ (1410)
Equation (1410) accounts for the relative attenuation of the fundamental
component which would be ignored if n H„ were chosen instead as the
figure of merit. When \K n \ = \H n \, the relative harmonic contents of the
input and output are equal. In any specific circuit the best signal, i.e.,
the most nearly sinusoidal, will appear at the output for \K„\ > \H n \ and
at the input for \K n \ < \H n \.
Phaseshift Network. One network commonly used in the socalled
"phaseshift oscillator," which we must therefore name "the phase
shift network," appears in Fig. 144a. Either the three series branches
(Zi) will be energystorage elements L or C and the shunt branches
(Zi) resistors, or vice versa. In general, all series branches and all
parallel branches are composed of equal elements; however, this is not
an essential condition. Satisfactory oscillators have been constructed
with widely unbalanced sections and with networks of more than three
sections.
Sec. 142] almost sinusoidal oscillations 437
From Fig. 144o, the network transfer characteristics can be found to
be
= ^ =
(1411)
e 3 (Z 1 /2 2 ) 8 + 5(Zx/Z 2 ) 2 + 6Zi/Z 2 + 1
Since either Z\ or Z%, but not both, is an energystorage element, the
Zx
Zx
Zi
1
:l
13
Z
2
z
2
z 2
^
<b) (c)
Pig. 144. (a) Basic "phaseshift network"; (6 and c) two specific examples.
oddpower terms in the denominator of Eq. (1411) contribute the
imaginary part of P(w). For it to vanish
(!)
" + 6!0
(1412)
In the specific case of the RC network of Fig. 1446, Z\ = 1/juC and
Zt = R; substituting these values into Eq. (1412) yields
&>o =
1
Vqcr
(1413)
As the elements in the network change, so will the form of the equation
defining coo
By either substituting the value of &> from Eq. (1413) intoEq. (1411)
or by making the simpler substitution of
(!)' = <
from Eq. (1412) into Eq. (1411) and by noting that the odd powers of
Z1/Z2 have vanished, the threshold value of /? becomes
0(«o) = Pi
V*
29
438
OSCILLATIONS
[Chap. 14
Thus the amplifier must have a minimum gain of —29 for sustained
oscillations. With this particular
type of network, the required gain
is independent of the elements com
prising Zi and Zi. Any other RC
or LR combination will give the
same value of gain but a different
operating frequency and network
input impedance. Since the neces
sary gain is negative, a single or an
odd number of stages are required.
Figure 145 illustrates one possible
circuit.
, for the RC network of Fig. 144&, ft,
A singletube phaseshift
Fig. 145.
oscillator.
In order to evaluate \H 2 \ and \K\
may be expressed as
1
1  30/n 2  j(Q V6/n)(l  1/n 2 )
where n represents the order of the harmonic. Taking A t = —29 and
substituting into Eqs. (148) and (1410) yields
\H*\
0.368
1.25
For this network, the waveshape containing the smallest amount of har
monics appears at the amplifier output. If the positions of R and C are
interchanged, this will no longer be true; the best signal would now appear
at the input to the base amplifier.
Variation of the oscillator frequency over the widest frequency range
requires the simultaneous adjustment of three similar elements. In the
network of Fig. 1446, any changes in jB will also change the driving
point impedance. This may load the base amplifier to a point where
oscillations can no longer be sustained. If C is varied, then we face the
problem of tracking three independent variable capacitors. Because of
these difficulties, the phaseshift network is generally used for a fixedfre
quency oscillator. Here only minor calibration adjustments are needed,
and they may be made by means of a single small trimmer capacitor or
padding resistor.
The Wien Bridge. The simple RC network used as an example in
Fig. 142 exhibits extremely poor selectivity. Any secondharmonic
components introduced are not only not reduced, but are actually
increased in amplitude. In this network \H t \  2.24 and \K 2 \ = 2.
But all is not yet lost. We can convert the simple network into a null
balance bridge (Fig. 146) and exchange necessary loop gain for selectiv
Sec. 142] almost sinusoidal oscillations 439
ity. In the bridge
e/ = et — ez
but et/d = of the simple network of Fig. 142 [Eq. (146)]. The
resistor combination Ri and Ri is ad
justed to satisfy the condition
ez = Ri = 1 _ 1
ei R t + Ri 3 8
Thus the overall 0' of the bridge of
Fig. 146 may be expressed as
18'
 £> 
ei
P
where
(H)
(1414)
1
3 + j(<»/(>>o — ojo/oj)
Fio. 146. A Wienbridge frequency
selective network.
The amplitude and phase response
of /S' are plotted in Fig. 1421. At the resonant frequency of the system
(coo = 1/CR), &' reduces to
P = T
and the threshold gain A t is equal to 8. The term 1/8 may be considered
as the degree of bridge unbalance; when 8 = », the bridge is perfectly
balanced at a> 0> and when 8 = 3, the circuit reduces to the simple network
of Fig. 142. If follows that the greater the degree of balance, the larger
the necessary loop gain to sustain oscillations. At perfect balance, an
infinite gain is required; with any physical amplifier, the circuit cannot
oscillate.
The improvement in selectivity may best be seen by substituting /3'
given by Eq. (1414) and A, = 8 into the expression for H n [Eq. (148)].
The result is
3 (1415)
H' =
8(1  3j3„)
where /3„ is the transmission factor of the RC network at the harmonic
frequency. Taking the ratio of H' n for the bridge to H„ found for the
completely unbalanced case (8 = 3) shows the dependence of the selec
tivity on 8.
H± _ 3/[8(l  3ft,)] _ 3
H n 1/(1  3ft.) 8
From Eq. (1416) we conclude that the percentage of second harmonics
(1416)
440
OSCILLATIONS
[Chap. 14
present at the output is reduced in proportion to the increase in required
amplifier gain. With a twostage amplifier having a gain of 300, the value
of \H' t \ is on ly 0.0224, a hundredfold reduction from the unbalanced case.
At the output of the bridge, i.e., at e s in Fig. 146, the improvement in
selectivity is not as impressive. K'„ may be expressed by
# , = 3  6(1  3ft.)
#"" 8(1 3A.)
K>
(1417)
As 5 gets large, the magnitude of Eq. (1417) approaches unity as a limit.
For other than very large gains, \K' n \ will always be slightly above unity.
To incorporate the Wien bridge into an oscillator, we need two iso
lated input terminals, usually the grid and cathode of the input tube as
shown in Fig. 147. Transformer coupling may also be used to convert
Fig. 147. A Wienbridge oscillator.
the doubleended network output to a singleended amplifier input.
With a transistor amplifier, a transformer would almost always be used
to avoid loading the bridge by the lowimpedance input.
In this circuit, we see that positive feedback for regeneration is supplied
through the RC branches to the grid of the input tube. The pure
resistive path introduces a negativefeedback voltage into the cathode.
The combination of both terms controls the operation of the circuit. At
wo the positive feedback predominates, and at the harmonics the net
negative feedback reduces the distortion components.
The basic circuit of Fig. 147 is used in many commercial widerange
audio oscillators of from 10 cps to 200 kc or even higher. To adjust
the frequency, usually both resistors are changed by steps and C is
changed smoothly as a fine control. We can also do the reverse — switch
C in steps and vary R continuously. Both arrangements perform
satisfactorily.
Other Null Networks. Almost any threeterminal null network may be
used as the frequencydetermining pair of branches in a bridge. The
other two arms are composed of resistors adjusted for the proper degree
of unbalance.
Sec. 142] almost sinusoidal oscillations 441
All arguments employed with respect to the Wien bridge will also
apply to these other networks, provided that care is taken as to the direc
tion of the unbalance. For some networks the greatest reduction in
harmonic content comes when 5 is positive, as in the Wien bridge, and for
others S must be negative. Depending on the network and the sign of
I — VW — t — WV — I
■1 — II — r II 1—9
(b)
Fig. 148. Two null networks, (a) BridgedT network; (6) twinT network.
unbalance at &jo, in some cases the resistive branches must be used for the
positive feedback, and in others the reactive branches so serve.
Figure 148 shows two networks which give zero transmission at their
null frequency. The feedback factors and further defining relationships
are tabulated in Table 141. A plot of the amplitude and phase response
Table 141
Network
Transfer function
Denning terms
BridgedT.
■
1j
Qo ta* — «o*
"' LC
r
2rC
TwinT.
■
1 j2
k + 1 Utifg
\/k OJ 2 — Id
1
2RR 2 C*
R o C
k = = 2 —
2Rt C 2
appears in Fig. 1421. At the oscillation frequency there cannot be any
feedback through the reactive elements and the necessary positive feed
back must be supplied through the resistive path. The basic circuit
configuration needed to satisfy this condition is shown in Fig. 149.
Here the value of /3' may be expressed as
P
Ri J R2
 P
T
442 oscillations [Chap. 14
and at the null point, the required threshold gain is
A t = 1 + Jl
As in the Wien bridge, when A, is large, the harmonic content of the out
put will be small.
To adjust the frequency of these two networks, we must vary three
elements simultaneously. The dif
ficulty that this entails makes
these networks best suited for fixed
frequency operation. For the same
degree of unbalance, the bridgedT
circuit will usually give the cleanest
waveshape. However, since it is
more difficult and expensive to ob
tain a highquality inductance than
Fig. 149 Oscillator connection when us  t ig tQ find extremely good resist ors
ing a null network for frequency deter . ■ • m i
mination. and capacitors, the twin1 network
might be preferred.
143. Transistor Feedback Oscillators. The relatively low base input
impedance of the transistor often prevents its direct substitution for
the highinputimpedance vacuum tube in the various oscillator circuits
previously discussed. Even after scaling down the resistors and increas
ing the capacity in the frequencydetermining network, the heavy load
ing makes it more difficult to establish the necessary unity loop gain.
For example, if the phaseshift network of Fig. 144 were connected
directly between the collector and base of a suitably biased transistor,
the output of this network would be heavily loaded by the input imped
ance of the transistor. Even though it is possible to sustain oscillations,
the frequency would be extremely dependent on the transistor parameters
—a relatively unsatisfactory situation. In order to solve this problem,
many transistor oscillators resort to some form of impedance matching
between the output of the feedback network and the input voltage
amplifier stage.
Figure 1410 shows a transistor phaseshift oscillator which employs
an emitter follower as the impedance transforming stage. It presents a
high input impedance across the output of the frequencydetermining
network and a low output impedance for coupling to the input of the
amplifier stage. By using this additional transistor, it now becomes
possible to make R « (1 + /3) R& and still keep the input impedance of the
phaseshift network large compared with the collector load R 3 of Ti.
The emitter follower has effectively isolated the elements controlling the
frequency from the remainder of the circuit.
Sec. 143]
ALMOST SINUSOIDAL OSCILLATIONS
443
Since at the frequency of oscillation the attenuation from e 2 to e x is
— M9. the base amplifier must have a minimum loaded gain (A = e*/ei)
of —29 to sustain the oscillation. Generally, the opencircuit (unloaded)
voltage gain of T t would have to be from 20 to 50 per cent higher.
Fig. 1410. A transistor phaseshift oscillator: T x is the voltage amplifier, and IT, an
emitter follower used for decoupling. The phaseshift network consists of three
identical RC sections where R a ^Rb = R
Fig. 1411. A transistor Wienbridge oscillator using a composite transistor input
stage.
A second method of increasing the effective input impedance is illus
trated by the Wienbridge oscillator of Fig. 1411. A composite transis
tor, which was discussed in Sec. 74, is used as the input stage of the base
amplifier. The two transistors Tj. and T t , taken together, have a baseto
emitter input impedance that is approximately (1 + /3) times as large
as that of a single transistor. Provided that reasonable values of
resistors are chosen for the bridge arms, the input characteristics of
the composite transistor will not adversely affect the selectivity of the
bridge. In order to decouple the bridge from the highimpedance
444
OSCILLATIONS
[Chap. 14
collector as well as to have a lowimpedance point from which to take the
output, we could also insert an emitter follower after TV All the results
of the discussion of the Wien bridge in Sec. 142 are directly applicable
to this circuit.
Currentcontrolled Oscillations. If the transistor is used in its natural
mode, with the appropriate form of feedback, then the oscillations pro
duced may be said to be current
controlled. The basic circuit con
figuration is shown in Fig. 1412.
Except that i replaces e, it is
identical with the voltagecon
trolled block diagram of Fig. 141.
To avoid confusion between the
voltage and currentfeedback fac
tors and with the forwardcurrent
amplification factor of the tran
sistor, we shall designate the
*l h+if
S
N
S
'/
.
'
r
Fig. 1412. Basic currentcontrolled oscil
lator: current amplifier A c , limiter L(i),
and frequencydetermining network T(a).
current feedback by T(u). This term is defined by
if = T(u)i 2
and as before, the complete frequency dependence of the system is
assigned to T(a>).
Fig. 1413. An example of a currentcontrolled oscillator.
The conditions for selfsustained oscillations (ii = 0) may be written
by analogy with Eq. (142).
1  T(f*)AMQ = ( 14 " 18 >
Equation (1418) states that the net loop current gain must be unity at
the frequency and amplitude of oscillations.
Figure 1413 illustrates a circuit that is the currentcontrolled equiv
alent of the simple oscillator of Fig. 142. By assuming that the load
Sec. 144]
ALMOST SINUSOIDAL OSCILLATIONS
445
impedance of the final stage is large compared with the input impedance
of the network, the full output current will flow into T(oi). If this
assumption is not justified, the current will divide proportionally between
Rl and the network and Rt will influence the oscillation frequency.
For the ideal case,
6 + J \oi/ wo — uo/w)
where wo = 1/RC. [Compare this equation with Eq. (146).] The fre
quency of oscillation is a>o, and the threshold current gain must be 3.
Fig. 1414. An oscillator using a tuned circuit as a selective network.
The only conditions which we must impose on the various RC and
RL networks, which may be used for feedback, are that T(w) must give
a clearly defined oscillation frequency and good harmonic rejection.
Also, to avoid coloring the network response, the associated current
amplifier should approach the ideal, i.e., a zero input impedance, a high
output impedance, and constant gain over the frequency range of interest.
144. Tunedcircuit Oscillators. At frequencies above 50 kc, it
becomes practical to use a highQ tuned circuit for frequency selection.
Because such a network is resistive only at its resonant frequency and is
reactive everywhere else, the oscillation condition of zero net phase shift
can be satisfied only at this particular frequency. Figure 1414 illus
trates how a portion of the developed voltage may be fed back to the
input of the base amplifier by tapping the tank circuit. Other methods
of feedback are also in use, and some of these will be discussed below.
Even if the excitation current is rich in harmonics, the low impedance
of the tank, everywhere but at its resonant frequency, will not permit the
development of other than an almost pure sinusoidal voltage. The feed
back will further improve the selectivity, and consequently we shall not
be particularly concerned with the harmonic response. We are, how
ever, quite interested in generating a signal whose frequency will not be
affected by external factors.
Almost all tunedcircuit oscillators may be represented by the basic
circuits of Fig. 1415o and b, where, in order to achieve the proper phase
relationship with a singlestage amplifier, the tap of the tuned circuit is
grounded. For simplicity, the biasing and power supplies have been
446
OSCILLATIONS
[Chap. 14
omitted. In any specific circuit these connections would depend on
the nature of the three impedance elements.
We might note that the network of Fig. 1415 includes two modes of
feedback, a direct transmission path from the plate (collector) to the grid
(base) through Z 3 and the mutual coupling linking the input and output
loops. Some oscillators use purely capacitive elements for Z\ and Z 2 ,
and in others two isolated coils are used. When we treat these cases, M
f
±
r
n
\
\
L /
"«.
!><
z>
z 2
Zx
Z 2
^■M"
^M^
z 3
z 3
(c
<)
(b)
>r p
~*
i— S— vvv
r d &A_ e
8
fCXM
'
\
e
h >
r >iN
e
(
Z x
V
Z z
Zi
z 2
<^3__
^AT^
>
C's
^~~M^
>
Z
3
(c) (d)
Fig. 1415. Basic tunedoscillator circuits, (a) Using a vacuum tube; (6) using a
transistor; (c) vacuumtube model; (d) transistor model.
would not appear. However, many of the more commonly used circuits
do contain coupled coils, and therefore M must be included in any general
treatment. Figure 1415 also shows the various models with which we
shall be concerned in the analysis of this type of oscillator. In the
interests of simplicity, the small reversevoltagetransmission term has
been omitted from the transistor model. If this term ever becomes
significant, it can always be included by inserting a voltage generator
h re v ce in series with r' n .
Since the single tube or transistor introduces an effective phase shift
of 180°, there must be an additional 180° contributed by the network,
composed of Z\, Zi, and Z 3 . These models may, of course, be treated
as before by evaluating A/3, where A is the gain when the output is loaded
by the input impedance of the feedback network and is the transmission
from the plate (collector) to the grid (base).
Sec. 144] almost sinusoidal oscillations 447
By observing that the criteria for selfsustained oscillation are inherent
in the vanishing of 1  0(u) A or 1  T(o>) A c in Eqs. (142) and (1418),
we can consider an alternative method of obtaining these criteria. The
nature of the overall response of any system may be found by solving
a set of mesh or node equations. This response is undefined when the
system determinant vanishes. Thus, by writing the loop equations, or
where appropriate the node equations, finding the system determinant,
and setting it equal to zero, we satisfy
1  0(a) A =0 or 1  T(w)A c =
The vanishing of the imaginary part of the determinant gives the fre
quency of oscillation. From the real part, we obtain the circuit condi
tions which must be satisfied if the oscillator is to be selfstarting.
Vacuumtube Tuned Oscillators. For the vacuumtube circuit of
Fig. 1415c, the two loop equations are
(r p + Zi)U — (Z 2 + Z m )i 3 = —iiei
 (Z 2 + Z m )i* + (Zi + Z 2 + Zz + 2Z m )i* = (1419)
One additional denning equation is needed before the set of equations in
(1419) can be solved. The control voltage must be expressed in terms
of the loop currents.
«x = Z m i 2 + (Z m + Zi)i, (1420)
Substituting Eq. (1420) into Eq. (1419) and collecting terms yields
the system determinant,
r p + Z 2 — y.Z m — (Z 2 + Z m — nZ m — p.Z%)
(Z, + Z m ) Z 1 + Z i + Z 3 + 2Z m
A =
(1421)
If we initially assume that the impedance elements are purely reactive
(Zi = jX t ), then the imaginary portion of the determinant will contain
only odd powers of Xi. The sign of the reactance is assigned to the ele
ment (plus for inductance and minus for capacity). It follows that the
frequency of oscillation will be given by
Im A = r„(Xi + X 2 + X 3 + 2X m ) = (1422)
Equation (1422) can be satisfied only when the sign associated with
one reactance is opposite to that associated with the other two. We
shall see below [Eq. (1423)] that Zi and Z 2 must be that same type of
element to satisfy the gain condition. Thus if Zi and Z 2 are inductive
at the oscillation frequency, Z 3 must be capacitive, and vice versa.
Since the sum of the loop reactances vanishes in Eq. (1422), the real
part of the determinant of Eq. (1421) may be expressed as
Re A = M (X, + X m )(X 2 + X m )  (X, + X.)»
448 oscillations [Chap. 14
By setting this equal to zero, we find that the threshold condition for
selfstarting is
M > y*tf" (1423)
Ai f A m
Equation (1423) can be satisfied only when Zx and Zi are the same type of
component: either both are capacitive or both are inductive. If they
are not the same, the feedback would be negative rather than positive.
Unless the ratio of impedances is somewhat less than /u, the overall
loop gain will not be greater than unity and the oscillator will not be self
starting. Furthermore, with threshold operation, the system is very
sensitive to changes in the parameters of the active element. On the
other hand, if this ratio becomes much less than m, the limiting may
introduce an excessive amount of harmonics, with a corresponding
deterioration of the waveform. To avoid both problems and to have
some latitude for component tolerances, at least one of the elements
determining the threshold point should be made adjustable.
The following possibilities arise where the major mode of feedback
is through X 3 and the mutual coupling either does not exist at all or is
purely incidental.
1. Colpitts oscillator (Fig. 1416o)
Xi and Xi are capacitors.
Xt is an inductance.
X m = 0.
2. Hartley oscillator (Fig. 1416&)
Xi + Xi + 2X m is a single tapped coil.
' X* is a capacitor.
3. Tunedplate tunedgrid oscillator (Fig. 1416c)
Xi and Xi are tuned circuits adjusted to be somewhat inductively
tuned slightly below a> .
Xt is the stray gridtoplate capacity.
X m = 0.
When the major mode of coupling is through the mutual inductance and
Xz represents the insignificant parasitic coupling, the following two cir
cuits are also feasible:
4. Tunedplate oscillator (Fig. 1416d!)
Xi is a tuned circuit in the plate loop.
Xi is the grid coil.
2X m is the mutual coupling between the two coils.
5. Tunedgrid oscillator
Virtually identical with the tunedplate oscillator except that the
tuned circuit is moved into the grid loop.
Sec. 144]
ALMOST SINUSOIDAL OSCILLATIONS
449
Various hybrid oscillators combining significant features of the five
listed above are also in use. The above tabulation only indicates the
range of possibilities; it makes no attempt to exhaust them.
(°)
(b)
'At
M
— L\\\ 1 ( ) C, i L 5 o
T
(c) (d)
Fig. 1416. Some practical vacuumtube tuned oscillators, (a) Colpitts circuit; (6)
Hartley circuit; (c) tunedplate tunedgrid; (d) tunedplate oscillator. The various
components not labeled are used for bias and decoupling and would be chosen to have
a negligible influence on the frequency of oscillation.
In the Colpitts oscillator of Fig. 1416a, the frequency of oscillation as
found from Eq. (1422) is
woe
where C T = C1C2/C1 + C 2 . For selfstarting, n > Ci/Cj [from Eq.
(1423)]. The Hartley circuit (Fig. 14166) oscillates at
J
1
and here
M>
Cz^ + U + 2ikf)
Li + M
Li + M
In both circuits the frequency of oscillation is simply the resonant fre
quency of the LC combination. The capacitive term in the Colpitts
450 oscillations [Chap. 14
equation is the series combination of C x and C 2 . If the tube has a
reasonably large gain, C 2 <C Ci and Ct = C 2 . In the Hartley circuit,
the inductive term is the total tank inductance and, with a highgain
tube, L2 » L\.
The Hartley and the two single tunedcircuit oscillators (circuits 2,
4, and 5 above) are best suited for variablefrequency operation. In
these, the frequency may be changed by varying a single capacitor.
Moreover, this adjustment will not affect the conditions for selfstarting
operation, which depends on the mutual coupling between two coils and
on the location of the coil tap. In the other oscillators either two LC
circuits must be retuned or two capacitors must be simultaneously
tracked.
If the tank elements are dissipative, or if they are loaded by the external
circuit when coupling to the next stage, then the associated resistance
will change both the frequency of oscillation and the conditions for
selfstarting. Provisions for retuning and for adjusting the amount of
feedback allow us to compensate for these changes. There is no point in
calculating the new condition for threshold operation. It is, however, of
some interest to recalculate the frequency of oscillation. By doing this
we shall see that the circuit dissipation also makes the frequency depend
ent on the parameters of the active element.
Example 141. Let us consider, for example, the basic Colpitts oscillator of Fig,
1416a, where the nominal unloaded frequency is w = 10 7 radians/sec. The tube
used has y. = 20 and r, = 5 K. The tank is so designed that Q \/L/Ct — 25 K and,
in addition, its loaded Q — 25.
From the above conditions, the tank components are found to be Ct = 100 ppf and
L = 100 iih. To ensure selfstarting and to allow for changes in the tube parameters
we shall satisfy
" = 2 C 2
instead of the threshold condition. Solving yields Ct = 110 npi and Ci = 1,100 ii/d.
The final parameter — the equivalent series coil resistance — is
10 7 L 10' X 10" .„ ,
r = .. = ^ = 40 ohms
By substituting Zz = r + jaL, Zi = —j/aiCi, and Z% = —j/aCz into Eq. (14211
and collecting the imaginary terms, the frequency of oscillation is found from
The frequency becomes
■7T0
"' \ LC,C 2 \ ' r Cy + Ct r v )
\ ^ 1,210 5,000/
Sec. 144] almost sinusoidal oscillations 451
or the frequency increases by approximately 0.36 per cent over its nominal value as a
direct result of the resistive loading of the tank. The conditions for selfstarting will
also change; this calculation is left as an exercise for the reader.
Unless r p S> r, the circuit would be quite sensitive to the changes in r p
occurring during the life of the tube or when replacing tubes. Any resis
tive loading of the tuned circuit would effectively increase r and would also
change the frequency of oscillation. External loading may be minimized
by using an amplifier to decouple the final load from the oscillator stage
and by loosely coupling this stage to the tank.
Transistor Tuned Oscillators. From the denning equations for the
transistor model of Fig. 1415d, the overall system determinant is found
to be
r' n + Z X Z m  (Z, + Z m )
A= pu + Z m n + Z t (Z* + Z m ) (1424)
(Zi + Z.) ~(Z» + Z m ) Zx + Z 2 + Z 3 + 2Z„
When the impedances are all purely reactive, the imaginary part of the
determinant will reduce to
v{,(Ii + X 2 + X 3 + 2X m )  X 3 (X 1 X 2  I« ! )  (1425)
where the reactance term includes an associated sign. We might note
that the first part of Eq. (1425) contains the total series impedance of
the selfresonant circuit composed of the three reactive elements. The
last portion is a correction term due to the nature of the external loading
of the tank. In general, we shall have to minimize the influence of these
external factors if the oscillator is to have good frequency stability.
The significant portion of the frequencydetermining equation is
identical with that found for the vacuumtube circuit. As discussed
above, the nature of Z 3 must be opposite to Z\ and Z 2 . Consequently
the possible circuit variations are those previously tabulated.
By assuming that the actual frequency of oscillation is very close to
the natural frequency of the tank, the term
X, + X 2 + Xz + 2X m £*
and the determinant may be simplified before solving for the condition
for selfstarting. It shall be found by setting the real part of the remain
ing terms equal to zero.
r' u (X 2 + X m y + r„(Xi + X m y  0n(Xx + X m )(X 2 + X m ) = (1426)
Because the second term on the lefthand side is relatively insignificant,
Eq. (1426) may be further simplified. The resulting criterion for self
starting oscillations becomes
A 2 f X m  pTd re_ f14— 97^
v i Y — 7' r' V 1 ^  *'^
■Al \ A m r u r n
452
OSCILLATIONS
[Chap. 14
To justify the use of this equation, we shall solve Eq. (1427) for (Xi +
X m ) and substitute back into the original expression.
rii(x, + x m y +
(r'nY
(X. + X„) 2  rJ 1 (X 2 + X.)« =
The neglected second term is only rJ 1 /j8V < j times as large as the first term.
Since r[ x is the small input impedance, ^Vj ^> r' n and the condition
expressed in Eq. (1427) is essentially correct.
(a) (b)
Fig. 1417. (a) Transistor Colpitts oscillator; (b) Hartley oscillator.
From Eqs. (1425) and (1427) the frequency and gain requirements
of the Colpitts oscillator of Fig. 1417o are
woe
\ LCxC
Cl . 1
where \Z(Ci + C 2 )/LCiCt is the resonant frequency of the unloaded
tank. For selfstarting, the Colpitts oscillator must be adjusted so that
£c > Ci
r'n ~ C 2
The Hartley circuit of Fig. 14166 oscillates at
1
UOH —
4
and for selfstarting,
C(Li + U + 2M)
r. ^ Lt + M
r' n Lt + M
L\Li
M*
rjru
In both cases the resistive loading of the tuned circuit causes a small
change in frequency from the nominal resonant point. This can be
Sec. 144] almost sinusoidal oscillations 453
minimized by maximizing the rjr'u product and by properly choosing
the ratio of L to C. Furthermore, by using tightly coupled coils in the
Hartley circuit, the numerator of the correction term will be reduced;
with unity coupling it will reach zero and the circuit will oscillate exactly
at o>o.
Crystal, Oscillators. A suitably clamped quartz crystal may be made to
resonate by exciting it with an electrical
signal of the proper frequency. The T
mechanical oscillations are fixed by the ■
relative dimensions of the crystal, and the 
damping depends on the characteristics of  — 
the mounting. As the damping factor can ~~ T~
be made quite small, the electromechanical I
Q will be proportionately large; expected I
values are between 1,000 and 50,000. To ^ 14 _ lg ^^^ electrical
use the crystal in a fixedfrequency oscil circuit of a quartz crystal,
lator, we simply substitute it for one of
the reactances in the basic tunedoscillator configuration of Fig. 1415.
Sinusoidal oscillation is possible at one of the two modes of electro
mechanical resonance corresponding to the electrical resonances of the
equivalent circuit of Fig. 1418. The parallelresonant mode depends on
the mass factor L t and the capacity of the mounting plates using the
crystal as a dielectric, C. At this frequency, the terminal impedance is
purely resistive and is extremely large. In its other mode, the mass and
spring constant of the crystal, L, and C,, are seriesresonant. Here the
crystal is essentially a short circuit. Since the dissipation term r, is
small, the network appears reactive everywhere but at these two fre
quencies. Many crystals are cut to have several resonant points, and
these might be incorporated into the electrical model by adding, in
parallel, additional LfiCi series arms.
Even though the frequency of oscillation is essentially that of the
crystal, minor adjustments are possible, without reducing the Q. By
adding series capacity or inductance the frequency can be changed by a
few parts in a million. For wider ranges, the variablefrequency circuits
would be used instead.
Two crystalcontrolled oscillators are shown in Fig. 1419. In the
Pierce circuit, the crystal controls the amount of feedback applied. At
its seriesresonant frequency, the low impedance presented increases
the amount of feedback to a point where oscillations can be sustained.
Since the grid circuit is capacitive, the crystal must be slightly inductive
for the proper phase of the feedback signal. To satisfy the conditions
found above, the plate tank would also be tuned somewhat capacitively.
To all other frequencies the crystal presents an extremely high reactance.
454 oscillations [Chap. 14
It thus reduces the loop gain well below the threshold value. Further
more the net phase shift of the feedback voltage will no longer be the
180° needed for oscillation.
The second circuit of Fig. 1419, the transistor oscillator, is a tuned
input tunedoutput circuit with capacitive feedback. Here the output
(a) (b)
Fig. 1419. (a) A Pierce oscillator using the crystal as a seriesresonant mode; (6)
a transistor oscillator where the crystal operates as a parallelresonant circuit.
MA
4HH
1megacycle
crystal
oscillator
Buffer
amplifier
and clipper
Differentiator
i
_
^
_
1kc
multi
10kc
multi
100kc
multi
)
100,«sec
period
10^sec
period
J,
1 1 1
lv<sec period
Fig. 1420. A crystaloscillatorcontrolled system designed for precise timing measure
ments.
will be developed when the crystal is a high impedance, i.e., at its parallel
resonant frequency. At all other frequencies the small reactance pre
sented will load the amplifier and will also cause the feedback voltage
to be in quadrature to that needed for oscillation.
Crystalcontrolled oscillators are widely used as frequency standards.
By amplifying, clipping, and finally differentiating a 1megacycle sinus
oid, we derive a train of negative pulses spaced 1 Msec apart. As shown
in Fig. 1420, these are next applied as a trigger source to synchronize
an astable multivibrator normally operating at 100 kc. The 10^sec
period pulse train may be further divided down by the other multi
Sec. 145] almost sinusoidal oscillations 455
vibrators shown. From this chain we derive pulse outputs spaced by
10 jusec, 100 jisec, 1 msec, etc. Since the base frequency can be main
tained to within 1 part in 10», this system has been used for highly
precise timing.
145. Frequency Stabilization. All oscillators are to some degree sus
ceptible to frequency changes caused by the resistive loading of the
selective network. If the loading were to remain constant, then the
initial calibration of the instrument would correct for the deviation from
the nominal tank resonance frequency. Unfortunately some of the dis
sipative elements may be expected to change with time, sometimes in one
direction and sometimes erratically. All the circuit components con
tribute to this drift; even slight changes in the powersupply voltage
will shift the operating point and thus affect the parameters of the
active element. Therefore any oscillator which is designed as a precise
frequency standard should be compensated with respect to all such
variations.
We implicitly assume that the nature of the elements comprising the
selective network is such that they themsebes are completely stable with
respect to aging and ambienttemperature variations.
Since it would be identical with detuning the tank, we are also neglect
ing the effects of any reactive loading of the tuned circuit. Some react
ance would always appear across a portion, or across the complete tuned
circuit, because of the interelectrode and stray capacity associated with
the tube and the transistor. Choosing a large tuning capacity would,
of course, make the frequency much less dependent on any changes in the
parasiticcircuit elements.
Our first objective, then, is to eliminate, as far as possible, any other
factors producing a frequency shift away from <o . Even though special
provisions may be made in each individual circuit, such as using tightly
coupled coils in the Hartley oscillator, in Sec. 144 we saw that the per
formance of all oscillators will be materially improved by minimizing the
external loading of the network. First, a buffer amplifier may be used to
decouple the oscillator from the varying output load, and second, since
the harmonics produced in the limiter are reflected as additional loading,
the circuit can be adjusted close to the threshold of oscillation. Finally,
we should investigate the mechanism of frequency instability with a view
toward selecting the circuit which best suits our particular needs.
Stabilisation Factor. The frequency of oscillation is identically that
frequency at which the net phase shift around the amplifier and feedback
loop is zero. Since the feedback network contains reactive elements, the
insertion of any resistance will change the phafie response and also the
null frequency. For example, assume that a change in one parameter,
external to the tuned circuit, produces a net phase shift of 5° at the
456 oscillations [Chap. 14
nominal frequency of oscillation /<>. For sustained oscillations, the
remainder of the circuit must contribute the complementary shift of
— 5°. If the normal rate of phase variation in the vicinity of / is — 10°
per 1,000 cycles, then a frequency increase of only 500 cycles will reestab
lish the condition of zero net phase shift. The new frequency of oscilla
tion would be /o + 500. From this simple example, we conclude that
the highest degree of stability would correspond to the most rapid change
of phase with frequency.
If, in an oscillator, we can find one set of elements whose phase varies
most rapidly with frequency, then these elements would have the most
pronounced effect on the overall frequency stability. This will, in gen
eral, be the frequencyselective network. As a measure of frequency
stability, we define the normalized sensitivity factor
S ' S A^To (14  28 >
The larger the value of S f , the more stable the oscillator with respect to all
changes in the circuit which may produce an undesirable frequency shift.
When, in the limit, S f becomes infinite, the frequency of oscillation is
completely independent of all other sections of the system.
Figure 1421o is a plot of the phase behavior of a Wien bridge. In the
vicinity of its null frequency, the response is similar to that seen in all
other unbalanced null networks (Fig. 14216). For this particular exam
ple, the phase of $', as found from Eq. (1414), is
(ii_o\_ tan _ 1 l/«_« \
\wo « / 3 \coo a /
This is plotted as a function of &>/&>o for three different degrees of unbal
ance. One of the three is the completely unbalanced bridge (8 = 3).
It is readily apparent that Sf reaches a maximum at &> and that the
closer the bridge is to balance, the larger its value. Differentiating Eq.
( 1429) with respect to to and multiplying by co yields
^ = tan J — <j —
Sf = Wo
3  8
d4 9__
da
1 +
( 3 — & \ 2 /<w _ woV ill (j± _ fjJoV
9 / \a>o «/ 9 \<«io w/
Since 8 > 3, both terms will have the same sign and each, will reach its
maximum at a = wo Evaluating S/ at this frequency,
S f . = %« (1430)
The minus sign indicates the direction of the phase change with frequency.
Sec. 145]
ALMOST SINUSOIDAL OSCILLATIONS
457
Equation (1430) shows that the bridge improves the circuit stability in
proportion to its degree of balance. We might note that previously we
had shown that this same condition was necessary for the optimum
harmonic rejection [Eq. (1416)].
1 1
8300
6
3
3 s
6=3
X. \, $
30
5300"*"^.
150"
120
90
60
30 ^.
8
I
30
60
90
120
150°
8 10
(a)
a
+90°
90'
Twin T and
Bridged T
(6)
\tf\
V3 for the Wien Bridge
1 for the Twin T and
Bridged T
(c)
Fio. 1421. Amplitude and phase characteristics of null networks, (a) Phase response
of the Wien bridge; (b) the phase response of unbalanced bridge using a twinT or a
bridgedT network for one pair of arms; (c) the general amplitude response of the null
networks.
Unless a sharp null is exhibited by the network, the phase changes
relatively slowly in the vicinity of « . The curve for S = 3 in Fig. 1421a
illustrates a variation of this nature. Since this curve represents the
behavior of a completely unbalanced bridge, it also typifies the response
of such EC and RL circuits as the phaseshift oscillator.
The phase response of a tunedcircuit oscillator is related to the
circulating current through the three reactive components Z\, Z t , and
458
OSCILLATIONS
[Chap. 14
Z%. Near a> the relative phase of the feedback voltage, which is devel
oped at the activeelements input, may be approximated as
4> = tan 1 Qo
(1431)
If this were to be plotted, we would have a family of curves similar to
those drawn for the Wien bridge. After all, a tuned circuit is also a
Fig. 1422. Meachambridge oscillator.
null network. The phase starts at 90°, reaches the 180° necessary for
the positive feedback at to , and then continues increasing toward 270° at
the higher frequencies. The frequencysensitivity factor of this phase
response, evaluated at o>o, is
5,. = 2Q
For optimum stability we should use a highQ circuit and should set
the oscillation frequency as close to co as possible. An extremely good
choice for one of the elements is a crystal which will give a frequency
stability factor of 10 s to 10 6 . With a normal tuned circuit, the best
that can be expected is about 10 2 .
For an extremely precise frequency standard we can combine the
improvement found in a highQ crystalcontrolled oscillator with the
selectivity multiplication of an almost balanced bridge. The Meacham
oscillator of Fig. 1422 has the highest stability of any circuit yet devised.
The bridge nulls at exactly the seriesresonant frequency of the crystal.
At this frequency the crystal appears completely resistive, which makes
the bridge easy to balance. By using a small degree of unbalance (1/5)
and approximately equal resistance arms, the stability factor will be
S f .  2SQ
and for Qo = 10 6 , stabilization up to 1 part in 10 8 is possible. This
means that if the amplifier introduces a phase shift of 0.1 radian (a
relatively large value), the frequency would change by only 1 part in 10 9
Sec. 145] almost sinusoidal oscillations 459
Reactive Stabilization. Llewellyn showed that it is possible to stabilize
an oscillator by inserting reactances in series with the terminals of the
active element or in series with the applied load. The theory behind
such stabilization is that the tank circuit appears resistive at its own
resonant frequency and that only the need to compensate for the addi
tional loop phase angles causes the frequency to shift. Now if the
external resistance sees a completely resistive internal impedance, it
would not affect the phase of the system; it would only increase the load
ing and the gain required for threshold operation. At many points in the
circuit the twoterminal impedance has a reactive part. When this is
loaded, the complete phase response of the system changes. By inserting
in series with the external load a reactance of the opposite type to that
originally seen, it is possible to tune the subsidiary loop and thus reduce
the net phase shift to zero. The load now sees a pure resistance, and the
frequency becomes independent of the load and of load variations.
The nature of this stabilizing reactance depends on the terminal at
which it is introduced and on the configuration of the overall network.
It is also possible, by inserting a single reactance at an appropriate point,
to compensate for the loading at several different parts of the network.
But in general, compensation elements would have to be inserted in
series with each of the resistive components of the external circuit.
As an example, we shall insert such a reactance in series with the
base in the Hartley transistor oscillator described in Sec. 144. For
simplicity, the mutual coupling will be taken as zero. Furthermore, since
the stabilization will return the frequency to cj , the series impedance
of the tank will also be zero. The simplified system determinant will
now be
r'„ + Z. + Z 1 Z 1
A =
Pn Z% + Ti —Zi
Zx z*
where Z, represents the stabilizing reactance. Setting the imaginary part
equal to zero (when all impedances are purely reactive) yields
X.Xf + X 1 X 2 i + X 1 i X i =
and this equation enables us to find the value of X,.
'•^O + fi)
But if the active element employed in the oscillator has a high gain,
then, from Eq. (1427), we see that 1 » Xi/Xt and the stabilizing react
ance
X. S  Zi (1432)
460 oscillations [Chap. 14
The element to be inserted in series with rj x is a capacitor whose reactance
at o) is given in Eq. (1432).
In order to obtain an order of magnitude for C„ consider a circuit
where C = 250 md, L x = 10 nh, L 2 = 990 jah, and « = 2 X 10 6 . Sub
stituting into the expression for X, yields
C ' = 5?Li = 4 X 10+ 12 X 10 X 10« = 0025 " f
This value is quite feasible, and it would appear in the circuit as an
input coupling capacitor. Note that C, depends on the frequency and on
the value of L\. For variablefrequency oscillators, stabilization of
this nature is obviously impractical.
146. Amplitude of Oscillations. The exponential buildup of the
selfstarting oscillator will continue as long as the average gain over
the cycle is greater than the threshold value. Eventually, the amplitude
of the output sinusoid is limited as its peaks force the instantaneous
operating point into the nonlinear regions. As a crude firstorder
approximation, we can say that the output will be equal to the linear
capabilities of the system. Since this depends on the nature and point of
application of the nonlinearities, we shall now examine several of the
amplitudelimiting schemes used in practical oscillations. Fortunately,
the exact signal level is much less significant than the exact frequency,
and therefore we can justify relatively gross approximations in treating
this problem.
1. Activeelement Limiting. The first case considered is where the
amplitude is limited by the tube (transistor) being driven into saturation
and/or cutoff. We shall make the following assumptions as to the
oscillator behavior.
1. The gain is adjusted to slightly beyond the threshold value so that
the output is not excessively distorted.
2. The selectivity of the frequencydetermining network will greatly
reduce the amplitude of the harmonics produced, and consequently there
will be an almost pure sinusoid present at one point at least.
Since the response is sinusoidal at one point in the closed system,
the oscillator behavior may be simulated by opening the loop at this point
and exciting the open circuit from an external sinusoidal source of the
same frequency. For example, in the Colpitts oscillator of Fig. 1423o,
the loop has been opened at the grid. To avoid changing the overall
response, the network must be loaded by the impedance it normally sees.
•The sinusoidal driving signal must also be injected through a source
impedance of the appropriate magnitude.
In our effort to describe the nonlinear behavior of the oscillator, with
out having to account for the different impedances presented by the tank
Sec. 146] almost sinusoidal oscillations 461
at its various harmonic frequencies, we shall assume that the plate load
is a constant equal to the resistance of the tank at resonance. This
incorrect assumption can be justified only by noting that, in an almost
sinusoidal oscillator, the fundamental component is the predominant
term. Any error resulting will be insignificant if the totality of the
d=C 2 Cn d= e f 0 o e b
(a)
AE.,
E P t
Slope. t;
E P i
~AE s i m
(b) (c)
Fig. 1423. (a) Opencircuit Colpitts oscillator; (6) platecircuit operating path at the
fundamental frequency; (c) plate waveshapes for two values of drive, assuming a
constantresistance plate load.
/ \
■M )
9\2 It 2¥
2n N
k <Oot
\ /
  ^_^
V.
harmonics calculated on the basis of a constantresistance load remains
small. But this is the same as saying that the net gain in the small
signal region should be adjusted to slightly beyond the threshold value.
By solving Fig. 1423 we find that the effective input resistance of the
unloaded highQ tuned circuit at resonance may be expressed as
B  = B = (crTc s ) 2< ^ (14  33)
From the selfstarting condition for this oscillator, C% < pCi, we see
that Ct is the smaller capacity; when /*• > 10, it will completely pre
462 oscillations [Chap. 14
dominate. Thus the nominal resistance of the tank at resonance reduces
to
R = Q \c;
Using the equivalent load resistance R, the path of operation of the
base amplifier is constructed (Fig. 14236) and the bounds of the linear
region delineated.
For an input excitation of
e, = E,im cos a>ot
the output remains sinusoidal up to the linear capabilities of the tube
(point x or y in Fig. 14236 and the corresponding point W in Fig. 1424).
Above this value the output is clipped by the nonlinearity as shown in
Fig. 1423c. In reality, the tank cannot sustain such a waveshape, but
from it we can determine the approximate amplitude of the fundamental
component of the tank voltage. Equation (1434) may be used:
1 /" 2ir
Epi™ = ~ I e p cos uot dwot (1434)
t Jo
where e p is the time varying component of the plate voltage. When the
output is not readily expressed in the form of an analytic function, the
fundamental will be found by a graphical integration or by a schedule
method.
Suppose, for instance, that the tube is biased so that the plate operating
path is symmetrical about Ebb One clipping point is at the E c = line,
while the other corresponds to cutoff. If we further assume that the
limiting in both regions is ideal, then the approximate waveshapes with
which we are concerned are shown in Fig. 1423c. From Fig. 14236,
the peak signal in the linear region of operation is
E p i = Ebb — Ebi = A(E B i m )i = ■= — , p Ebb (1435)
where A = —nR/(r p + R). In a transistor oscillator biased in the
center of its active region E p i = Ebb
Because of the symmetry exhibited by the signal of Fig. 1423c,
Eq. (1434) reduces to
4 /"» 4 f r/2
Epi m =  / E P i cos <i>ot dwot +  I AE. lm cos 2 wot daat
ir Jo r Ja
=  E Pl [sin 6 + ^^ (2 2 ) J (14 " 36)
where 6 = cos 1 — ~
AE,
Sec. 146]
ALMOST SINUSOIDAL OSCILLATIONS
463
The function describing the amplitude response of the oscillator, that
is, E p i„ versus E, Xm , may now be plotted (curve a in Fig. 1424). This
same describing function is often obtained experimentally, by measuring
the output as the input excitation is increased. At each point on the
curve the ratio of E pim /E, lm is the average gain of the base amplifier over
1 cycle of the fundamental. Since the criterion for the stable oscillation
is satisfied when the gain is equal to 1/jS , by superimposing a line having
Fia. 1424. Curve a, describing function for the oscillator of Fig. 1423; curve 6,
describing function of a nonselfstarting oscillator.
this slope and by locating its intersection with the describing function, we
find the amplitude satisfying
Av —
E.
E,\
J.
00
When the smallsignal gain is only slightly greater than the threshold
value, point Z will fall close to the upper limit of the linear range of the
circuit. Thus the area between the l//3 line and the describing func
tion gives a qualitative measure of the harmonic content of the output;
if the area is small, the signal will be an acceptable sinusoid; if it is large,
the output will be somewhat distorted. That Z of Fig. 1424 is the sole
stable point may be verified by considering the response to a small
change in the input excitation. If the input is momentarily reduced,
then the new value of loop gain becomes greater than unity, indicating
that the signal must now increase with time. Above point Z, the average
loop gain is less than the minimum value needed to sustain the oscilla
tions and they begin decreasing toward the stable value.
Curve b in Fig. 1424 illustrates the shape of a describing function of
a nonselfstarting oscillator. This might be the circuit of Fig. 1423
when the tube is biased on the verge of cutoff instead of in the center of
its linear region. At smallsignal levels the average gain is less than 1//S
464
OSCILLATIONS
[Chap. 14
and the circuit is unable to sustain oscillations. However, if some
external excitation forces the operating point past S, the oscillatory
criterion is satisfied and the amplitude will continue to increase toward
point Z'.
, .„ AE, lm cos8,.~
(a)
Fig. 1425. (o) Gridbiaslimited Colpitts oscillator; (6) plate waveshapes for small
a and large 6 excitation; (c) describing function for a circuit with bias limiting.
Nonsymmetrical clipping of the sinusoid will produce an additional
dc component in the plate current. It would be found from
2t Jo
R
dut
and since 7 d0 flows through R K , the bias would change slightly. The
resultant shift in the transfer characteristics might have to be considered
when constructing the describing function. This shift is in the direction
of more symmetrical operation, and it therefore aids in reducing the
harmonics present in the output.
2. Bias Limiting. The general circuit response is similar to that
considered in case 1, except that the bias is derived by the grid current
charging of C„ (Fig. 1425a). The only difference in the analysis is that
Sec. 146] almost sinusoidal oscillations 465
the input sinusoid would be clamped at zero by the energystorage ele
ment included in the input lead. Consequently the position of the load
line shifts with the drivingsignal amplitude. The locus of the minimum
plate voltage is the E„ = line. As illustrated in Fig. 14256, until such
time as the grid is driven below cutoff, the output voltage will vary sym
metrically about E».
^*^4fc^4p^
Fig. 1426. Intermittent oscillation in a gridbiaslimited oscillator.
E P i m is readily evaluated once 6 is defined. From the clamped grid
circuit waveshape of Fig. 14256,
e c = —E.i m + E. lm cos &xrf
Hence is given by
, = cos _ 1 ( 1+  f J
_ E» + AE, lm cos 9
where &«, — ~
/*
For this type of limiting, the amplitude would also be found from the
describing function of the system (Fig. 1425c). Of particular interest
is that the output can reach a maximum and then decrease with an
increasing input. When the output amplitude is stabilized on the
. decreasing segment of the describing function, the circuit is prone to
intermittent oscillations, or "squegging." On the positive peaks the
large pulses of grid current charge the input capacitor. If the net charge
accumulated in C„ on the positive peaks is greater than the energy
dissipated in B s over the remaining portion of the sinusoid, the bias
voltage will build up to a point where it cuts off the tube. The larger
the net charge accumulating, the fewer the number of cycles contained
in the oscillation interval. As the oscillations damp out (Fig. 1426),
C recovers toward zero, with the long time constant C g R g . The tube
eventually turns on, and the cycle repeats. Reducing the time constant
will often allow the circuit to oscillate properly.
3. Feedback Limiting. In this case, a passive nonlinear element is
included in the feedback network at a point where it can change the
magnitude of (3 but where it will not affect the null frequency. For
466
OSCILLATIONS
[Chap. 14
Fig. 1427. Amplitudecontrolled Wien
bridge oscillator.
example, in the Wienbridge oscillator of Fig. 1427, the degree of bridge
unbalance is controlled by the thermal characteristics of the tungsten
lamp, which is used as one of the resistive arms. Its resistance increases
with temperature, and hence with the voltage applied across the branch
and across the bridge (Fig. 1428). Normally the bridge is slightly
unbalanced (0' = 1/8), adjusted so that the amplifier can just sustain
the given amplitude of oscillation.
Any increase in the output voltage
will increase the resistance of the
lamp and bring the bridge into
closer balance, thus increasing S.
The loop gain will drop below unity,
and the oscillations will start damp
ing out. Any decrease in the out
put would establish the conditions
necessary for an amplitude buildup.
We conclude that this form of re
sistive limiting maintains a con
stant output amplitude with a high degree of stability. The normal
operating condition is now given by
A0'(e,u) = 1 (1437)
where A is simply the gain constant of the system. At o> this reduces to
S(e)
and the particular voltage e satisfying this equation is the oscillation
amplitude. Since the limiting process readjusts the /?' of the network,
this oscillator will always operate under the threshold condition expressed
in Eq. (1437). Moreover, as the amplifier is never driven into its
nonlinear region, the output will be almost completely free of harmonic
distortion.
The lamp has thermal inertia, and consequently it will only respond to
the mean value of the voltage applied over some small time interval. A
standard small bulb, operating in the range of 600 to 1000°K, far below
the temperature used for illumination, has a thermal time constant
between 20 and 75 msec. The mass of the filament and the construc
tion of the bulb are the determining factors; the smaller the nominal
power rating of the lamp, the shorter the time constant. With respect
to any single cycle, the tungsten lamp may be treated as a fixed resistance.
At the lower audio frequencies where the period is comparable to the
thermal time constant (below 20 cps), the change in resistance over the
Sec. 146]
ALMOST SINUSOIDAL OSCILLATIONS
467
individual cycle distorts the output sinusoid and sets a lower frequency
limit for this method of amplitude limiting.
1,200
1,100
1,000
R„ *°°
40 800
35 700
30 600
25 500
20 400
15 300
10 200
100
^ Mazda
//
f
6w
120 v
//
/s
«400+45.
;
//
V
•''
6
>
■z^*,
6..
150
pilot
Jv
ma
amp
//
^s'
/ i
/£■
<^
■KM
14£
''/
2 4 6 8 10 12 14 16 18 20 E a — *
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 E b *■
E rms , volts
Fig. 1428. Resistance characteristics of two typical tungsten lamps and their linear
approximations.
Example 142. Let us consider the design of the amplitudelimited bridge of
Fig. 14t27. The circuit contains an amplifier which has a nominal gain of 300 and a
dynamic range of 120 volts peak to peak. The 6watt Mazda lamp of Fig. 14^28 will
be used for amplitude limiting.
To allow some latitude for variations in the amplifier response, we shall limit the
output sinusoid to 30 volts rms. With the large gain available, the bridge will be
almost completely balanced. For threshold operation Si = At — 300, leading to
«.
Ri + i?2
300
= 0.330
The voltage across the lamp is
Est  0.33S.  9.9 volts
Since a high degree of quality .control cannot be expected in a device made for
illumination and not originally designed as a control element, the actual lamp charac
teristics will vary over very wide limits. Any reasonable approximation will serve
for design purposes; for simplicity we shall use a straight line. Within the range of
468 oscillations [Chap. 14
interest (2 < E < 18), the Mazda lamp will have a resistance variation given by
R, Si 400 + i5E
At 9.9 volts across the lamp, the effective resistance is 845 ohms. Substituting into
the bridge arm equation,
fli
1.0  0.33
0.33
Rt = 1,706 ohms
Ri would be made somewhat adjustable to allow for variation in the lamp resistance.
We shall now assume that the bridge is balanced as calculated above and that the
amplifier gain decreases by 33 per cent because of the aging of the tubes. To sus
tain the oscillations, R 2 must also change until it satisfies
R,
fl„ + fti
200
= 0.3283
Thus Ri becomes
«, = (! 0.0076)B 2 = 838.6 ohms
The decrease of 0.76 per cent in the lamp resistance corresponds to a new lamp volt
age of
E K ,
9.755 volts
The new stable output is 29.71 volts. A gain reduction of 33 per cent is accompanied
by an amplitude change of only 1.0 per cent. Consequently, this method of limiting
also ensures amplitude stability. At the new output, the oscillator again operates
under threshold conditions.
The technique of limiting by automatically controlling the feedback
has also been applied to the Meacham bridge (Fig. 1422), where the lamp
replaces R$, and to several tuned
circuit oscillators, where a resistance
bridge containing lamps might be
used for feedback. Figure 1429
illustrates an amplitudecontrolled
bridge used to separate the feedback
control from the frequencyselective
network. Limiting may be as
signed solely to the bridge. Other
tuned circuits have used small lamps
in series with the coil to control the
effective Q of the tank.
Fig. 1429. Bridgecontrolled feedback
oscillator.
Besides tungsten lamps, most semiconductor materials (e.g., carbon,
silicon, and germanium) and many compounds, such as silicon carbide,
exhibit temperaturesensitive resistance properties. All these have been
used for amplitude limiting. But since the semiconductor thermistors
have a negative temperature coefficient of resistance, in contrast to the
positive one of the pure metals, they will be used to replace the comple
mentary arm of the bridge. In the Wienbridge circuit of Fig. 1427, the
ALMOST SINUSOIDAL OSCILLATIONS
469
Sec. 146]
thermistor would substitute for Ri instead of Ri. Thermistors are
fabricated in a much wider range of resistance, and rate of resistance
variation with temperature, than can be expected from a tungsten
filament. Operating as they do at a lower temperature (300 to 375°K),
they are extremely sensitive to changes in the ambient. For acceptable
amplitude stability some form of temperature compensation would be
necessary.
4. Automatic Gain Control. Threshold operation of the oscillator
may also be ensured by using the output amplitude to control the gain
of the base amplifier. With many active elements it is possible to find
some dc voltage or current which
will determine the magnitude of the
forward transmission. For exam
ple, in a pentode the gridtoplate
transconductance is a function of
the suppressor voltage. Remote
control tubes are designed with a
gain that decreases with the control
grid bias. In a tetrode junction
transistor the value of /3 may be
adjusted within relatively wide
limits by injecting a bias current
into the second base connection.
^*%ny
Fig. 1430. Basic automaticgaincontrol
oscillator — Hartley circuit.
And in all tubes and transistors, the
smallsignal gain varies widely with the quiescent point when operating
close to cutoff.
By deriving the gaincontrol voltage from the oscillator output, any
increase in the amplitude will create the conditions that will automatically
reduce the gain and the signal. The response to a decrease in the output
would be just the opposite. This action is similar to the thermally
adjusted bridge previously discussed. To avoid distorting the individual
cycle, the sinusoidal output is rectified and applied through a relatively
long time constant to the gaincontrol element. Therefore the amplitude
defining equation may be written
A(E)h = 1
where E is the dc control signal.
Consider the general problem posed in Fig. 1430, where the control
terminal is indicated by 7. The output of the oscillator is coupled
through the extra coil winding, rectified, and applied to this terminal.
As shown, the gain is assumed to decrease with any increase in E y (or
with an increase in the oscillation amplitude). For example, the linear
approximation of the transconductance may be expressed as
g n = <M  KE y (1438)
470 oscillations [Chap. 14
where K is a positive constant. If g n were to increase with E y , then
the control loop would be regenerative instead of degenerative and the
output amplitude would increase until the active element limits.
At any particular amplitude, the threshold of oscillation, as given by
Eq. (1423), is
Li + M
Taking the various coupling factors and amplitude transformations into
<*) (b)
Fig. 1431. Two amplitudecontrolled oscillators, (a) Junction tetrode transistor
circuit — Hartley oscillator. The control current is injected into the second base
(0 = 0o — Kit,). (6) A pentode tunedplate oscillator where the suppressor is used
to control the effective transconductance.
account, this condition may be expressed as
gmffp — K'Eq =
U + M
U + M
E is the peak value of the sinusoidal output and will be found from the
solution of this equation.
Figure 1431 shows two oscillators using automatic gain control to
maintain the threshold operation. The first one is a transistor oscillator
where the gaincontrol current is injected into the second base connection.
Variations in over a range of 20:1 have been measured for control
current variations between and 2 ma. In the second circuit, the control
signal is fed to the suppressor. An alternative arrangement, where the
roles of the suppressor and the control grid are interchanged, has also
been used; the greater effectiveness of the control grid in varying the
gain of the tube would further improve the amplitude stability.
In all the amplitudecontrolled circuits, the variation of the con
trolled source with respect to the variable bias is difficult to express in
Sec?. 147] almost sinusoidal oscillations 471
general terms and will have to be evaluated in each individual case.
Because the range of operation is severely restricted, a few measurements
are usually sufficient. The best region in which to operate is the one
where the variation of the controlling element with voltage is most pro
nounced. Even though satisfactory limiting has been obtained by this
method, the selectivity multiplication of the bridge makes it superior
when the output amplitude must be maintained constant. For an
extremely high degree of stability, both methods of limiting may be
incorporated in a single circuit.
g ml  3,000 /anhos
^^2,000/imhos
Eplm
E.l~
Fig. 1432. Amplitude stability illustrated by describing functions.
147. Amplitude Stability. As a second figure of merit for the oscil
lator, we might define a normalized amplitude stability factor
„ x _ dx/x
°* ~ dE/E
(1439)
where x is the variable having the most pronounced effect on the ampli
tude. We interpret S% as meaning that a 10 per cent change in x pro
duces only a 10/S% per cent change in the amplitude of oscillation.
In those circuits where the limiting depends on the nonlinearity of the
active elements, S% may be found graphically by plotting a family of
describing functions for various values of x. This process is illustrated
in Fig. 1432, where the variable of interest is the g m of the tube. At the
intersection with the 1/00 line, the stability factor would be given by
Oft. r^ Agm/ffm.T _ (gml
gm2)/(gwl + gml)
AE/E„ (E x  E 2 )/(Ex + E t )
For the values shown in Fig. 1432,
SJp = 3.8
If the amount of feedback were to be increased, then the stable point of
472 oscillations [Chap. 14
operation would fall on the more nearly horizontal portion of the describ
ing function and the stability would improve. However, under these
conditions, the harmonic content of the output would also increase, and
as this represents a loading on the frequencyselective network, the
frequency stability would be adversely affected.
In the automatically stabilized circuits, such as the Wienbridge
oscillator of Fig. 1427, any change in the base gain is compensated for
by controlling the degree of bridge unbalance. Since A is always equal
to d, the stability factor of interest is S S E . For the Wien bridge
Ro + KE 11
R a + KE + Ri 3 5
By taking the differential, there results
(1440)
(J?i + flo + KE) 2 S 2
But from Eq. (1440),
RlK  2 dE = ±d6 (1441)
Rl 2 +\ (1442)
Ri + Ro + KE 3 6
Since S » 1, the result of substituting Eq. (1442) into Eq. (1441) and
of solving for the stability factor is
S ^I & R^TKE VW
As the rate of the resistance variation with voltage K increases, S%
will approach 28/3 as an upper limit. Thus the amplitude stability is
directly proportional to the amplifier gain and to the degree of bridge
unbalance. Maximizing Eq. (1443) by using a highgain base amplifier
also optimizes the frequency stability [Eq. (1430)] and the harmonic
rejection ratio [Eq. (1416)].
For the particular values used in Example 142, S = 300, the amplitude
stabilization becomes
S E = 35.3
This is many times the stability that could be obtained in the pureresist
ance bridge unless the signal is distorted beyond recognition by the
limiting in the tube.
The automaticgaincontrol circuit (Fig. 1430) has a stability factor
directly proportional to the rate of change of g m with the control voltage.
Since
gm =* £Uo — K'Ey
ym
For optimum stability a tube whose element has the most pronounced
control over the transconductance should be chosen. Moreover, if the
ALMOST SINUSOIDAL OSCILLATIONS
473
signal were to be amplified before applying it for control, the stability
factor could be increased to almost any desired value.
PROBLEMS
141. (a) Find the frequency and necessary gain for sustained oscillation if the
phaseshift network of Fig. 145 consists bf four equal RC sections instead of three.
(6) Evaluate \H*\ and \Ki\ for this network. Compare the answers with the
harmonic coefficients found for the threesection network.
(c) Can the conditions for oscillation be satisfied with a twosection network?
Explain your answer.
142. (a) The network of Fig. 144c is used in the phaseshift circuit. Calculate
the necessary amplifier gain, the frequency of oscillation, and the two harmonic factors.
(6) Repeat part o if R and L are interchanged.
143. The frequency of the phaseshift oscillator of Fig. 145 is to be controlled by
varying only one of the three series capacitors over the range 0.1C < Co < IOC In
which section should this capacity be placed if the range of frequency variation is to
be maximized? What are the two bounding values of to, expressed in terms of
o>, = 1/(V6«0?
144. Calculate the input impedance of the three/SCsection phaseshift network
at «o Under what conditions will it be invariant as the null frequency is changed?
What is the oscillation frequency when the output impedance of the amplifier is equal
to ij? What is the new value of the threshold gain?
145. Verify Eqs. (1415) and (1417). Find H„ and K n for the Wien bridge for
2 < n < 5. (Let S = 300.)
146. The base amplifier of Fig. 147 has an unloaded gain of 200 and an output
impedance of 5,000 ohms. It employs, for the input stage, a tube having y. = 20,
r„ = 10 K, and a plate load of 5 K. If wo = 10* and fls = 10 K, specify the remain
ing parameters of the bridge. What is the actual value of 5? Make all reasonable
approximations in the course of your solution and take into account the loading of the
amplifier by the bridge and the loading of the bridge by the amplifier.
147. Repeat Prob. 146 when the bridge is decoupled from the output stage of the
base amplifier by means of a cathode follower. Its output impedance should be taken
as 400 ohms.
148. Calculate the oscillation frequency of the circuit of Fig. 1413 when Rl 2> R
What are the harmonicrejection factors? Plot the locus of the system's poles as A c
increases from zero.
149. In the circuit of Fig. 1433, what is the minimum value of for sustained
oscillations? At what frequency does this circuit operate?
?+30v
Fig. 1433
474
OSCILLATIONS
[Chap. 14
1410. A vacuumtube tunedplate oscillator (Fig. 1416d) has a plate tank circuit
consisting of a 10mh coil and 100/x/af capacitor. The grid coil is 5 mh, and the mutual
inductance is only 1 mh.
(o) Specify the frequency and the minimum value of n for oscillations.
(6) If the plate tank circuit has Qo = 50 and if r p «• 5 K, how would the frequency
change? Under this condition, what is the minimum value of g m needed for proper
operation?
1411. Consider the following two cases of the general feedback problem posed in
Fig. 1415a. All elements are assumed to be purely reactive.
(a) When there is no mutual coupling between Xi and Xj.
(6) When there is mutual coupling but Zs is an open circuit.
Under these conditions, calculate separately the gain and the feedback factor and
solve for the frequency of oscillation. If a high/* tube is used, what simplifications
may be made in evaluating /3 for the Colpitts and for the tunedplate oscillator?
1412. A tube having n = 20 and r, = 10 K is used in a Hartley oscillator where
the two coils are completely isolated (M =0). It is to be tuned over a range of 500
to 1,500 kc by using a capacitor that varies from 300 nid down to 30 nnl . If the Q of
the complete coil is 20 at 1 megacycle, specify the two inductances that will permit
oscillation over the complete range of frequencies. Be svue to check the end points
and to account for the loading effect of the tank resistance.
1413. Verify Eqs. (1425) and (1427) and the frequency and selfstarting condi
tions given in the text for the Hartley and Colpitts transistor oscillators.
1414. A transistor used for a Colpitts oscillator has r n = 500, r c — 1 megohm,
and f) = 50. The nominal frequency of oscillation is too = 10'. Plot the deviation
from wo as a function of the tank impedance. Assume Qo remains constant. Take a
range of tank impedances from 1 K to 1 megohm. Under what conditions will the
frequency of oscillation be least susceptible to changes in the external loading?
1416. Figure 1434 shows a threephase transistor oscillator where the three
secondaries are symmetrically loaded.
(a) Calculate the frequency of oscillation, assuming threshold operation.
(b) What is the maximum load that can be applied at the secondary without caus
ing the cessation of oscillation ?
(c) Calculate the frequencystability factor and the harmonicrejection factor \H t \.
(3=50
r{ i500
r c lM
»10:l
Fig. 1434
ALMOST SINUSOIDAL OSCILLATIONS 475
1416. Compare the frequencystability factor of the phaseshift oscillator of Figs.
145 and 1410 when the frequencydetermining network consists of three sections to
one where four RC sections are used.
1417. (a) Verify the frequencystability factor for the Meacham bridge.
(6) Two identical crystals are used in this bridge instead of one. The second one
replaces flj in Fig. 1422. If each has Q = 10 6 and the degree of bridge unbalance is
a = 100, what is the new frequencystability factor? The nominal resonant fre
quency of the crystal is 10' cps, and the change in the amplifier phase shift may reach
a maximum of 10°; what is the effective change in frequency?
1418. Find the frequencystability factor for the transistor currentcontrolled
oscillator of Fig. 1433.
1419. A Hartley oscillator is adjusted for threshold operation at/o = 1 megacycle,
using C = 250 jipf. As the g m of the active element changes, the tap on the coil is
accordingly changed to maintain the oscillations. Assume that there is 10 /»/if of
stray capacity from the input to the output of the active element; we wish to mini
mize the effect of the stray reactance on the frequency of oscillation. Should a
device with a low or high value of transconductance be used? Explain your answer
and justify it fully by comparing two circuits, one which uses a tube having n = 100
and r, = 50 K and the other a transistor where r' u = 500 ohms, /3 = 100, and n «
100 K. Which device would make a more stable oscillator?
1420. A transistor used in a Colpitts oscillator has the following parameters:
r u = 20, r c = 2 megohms, and a = 0.98. The oscillator uses a 50mh coil to set the
nominal frequency of u — 5 X 10" radians/sec. We can expect stray capacity of
5 ju/*f between the base and collector connections.
(a) If the circuit is initially adjusted so that the loop gain is twice the threshold
value, by how much will the frequency shift as a result of the stray capacity?
(6) Repeat part o for a transistor having a = 0.99. The nominal gain is again
twice the threshold value.
(c) If the tank has a loaded Q = 20 and if the phase of the base amplifier is 10°, by
how many cycles would the frequency shift from w ? Compare the answer with that
obtained in parts a and b.
1421. The tuned circuit in the Colpitts oscillator of Prob. 1420 has a Q of 20.
Without accounting for the effects of the stray capacity, calculate the reactance
which would have to be inserted into the base lead in order to return the frequency of
oscillation to wo. Repeat the calculations for a stabilizing reactance placed in series
with the emitter.
1422. Would it be feasible to minimize the effect of stray capacity by means of a
stabilizing reactance? Explain and justify your answer by considering the vacuum
tube Hartley circuit of Fig. 1416b.
1423. Given the Colpitts oscillator of Fig. 1423 adjusted to 25 per cent above
the threshold value of gain. The parameters of interest are Ebb = 150 volts, r, —
20 K, /i = 50. The tank circuit has a loaded Q = 20 and an input impedance of 30 K
at its resonant frequency of 10 6 cps. Bt is chosen to give a symmetrical transfer
characteristic.
Verify Eqs. (1435) and (1436).
Plot the describing function by first finding the maximum value of the fundamental
component of plate voltage in the linear region; next the output when the cosinusoidal
signal is clipped from —15 to +15°; and then when it is clipped from —30 to +30°;
etc. Represent the describing function by a sequence of straight lines connecting
these points. Find the approximate amplitude of oscillation. Calculate <SJ at the
operating point.
476 oscillations [Chap. 14
1424. Repeat Prob. 1423 when the limiting is controlled by the assumed perfect
grid circuit clamping.
1426. A transistor tunedcollector, tunedbase oscillator, operating at 10 6 cps, has
an effective collectorcircuit tank impedance of 5 K. Em, = 20 volts, r' u = 100, and
= 100. The circuit is adjusted to 50 per cent beyond the threshold. The bias is
derived by assumed ideal basecircuit clamping.
(a) Approximate and plot the describing function and find the amplitude of
oscillation.
(6) Evaluate Sj,™ at the operating point.
1426. Design an amplituderegulated Meachamb ridge transistor oscillator using
the 6.3volt pilot lamp of Fig. 1428 as the control element. The peak amplitude
across the bridge should be 3 volts. The crystal, at its seriesresonant frequency,
may be replaced by a 20ohm resistor. Specify all the elements of the bridge and
draw the basic amplifier indicating the various elements used for impedance matching.
1427. Find the output amplitude of a Wienbridge oscillator where the bridge is
initially unbalanced by a factor $ = 100 and where the amplitudecontrol element is
a thermistor having the approximate characteristics R = 10,000 — 25QE ohms. The
associated pureresistance bridge arm is 3,000 ohms.
1428. Repeat Prob. 1427 if the remaining resistance arm is replaced by a tungsten
lamp having the approximate characteristics R' = 1,000 4 100E. Thus two tem
peraturedependent elements are used in the bridge. Evaluate S S E .
1429. Evaluate the amplitudestability factor of the Meacham bridge of Prob.
1426. What combination of arm resistances would maximize this factor? (The
bulb may be changed if necessary.)
BIBLIOGRAPHY
Anderson, F. B.: Sevenleague Oscillator, Proc. IRE, vol. 39, no. 8, pp. 881890, 1951.
Bode, H. W.: "Network Analysis and Feedback Amplifier Design," D. Van Nostrand
Company, Inc., Princeton, N.J., 1945.
Bollman, J. EL, and J. G. Kreer, Jr. : Application of Thermistors to Control Networks,
Proc. IRE, vol. 38, no. 1, pp. 2026, 1950.
Bothwell, F. E. : Nyquist Diagrams and the RouthHurwitz Stability Criterion, Proc.
IRE, vol. 38, no. 11, pp. 13451348, 1950.
Chance, B., et al.: "Waveforms," Massachusetts Institute of Technology Radiation
Laboratory Series, vol. 19, McGrawHill Book Company, Inc., New York, 1949.
Edson, W. A.: "Vacuumtube Oscillators," John Wiley & Sons, Inc., New York,
1953.
Ginzton, E. L., and L. M. Hollingsworth: Phaseshift Oscillators, Proc. IRE, vol. 29,
no. 1, pp. 4349, 1941; also corrections, vol. 32, no. 10, p. 641, 1944.
Hooper, D. E., and A. E. Jackets: Current Derived ResistanceCapacitance Oscil
lators Using Junction Transistors, Electronic Eng., vol. 28, pp. 333337, August,
1956.
Keonjian, E. : Variable Frequency Transistor Oscillators, Elec. Eng., vol. 74, no. 8,
pp. 672675, 1955.
Llewellyn, F. B. : Constantfrequency Oscillators, Proc. IRE, vol. 19, no. 12, pp. 2063
2094, 1931.
Meacham, L. A.: The Bridge Stabilized Oscillator, Bell System Tech. J., vol. 17,
pp. 574590, 1938; also Proc. IRE, vol. 26, no. 10, pp. 12781294, 1938.
Nyquist, H.: Regeneration Theory, Bell System Tech. J., vol. 11, pp. 126147, 1932.
CHAPTER 15
NEGATIVERESISTANCE OSCILLATORS
The energy needed to sustain oscillations can be supplied by shunting
the dissipative tuned network with the negative drivingpoint resistance
of an active element. The solution of this problem is well defined, and
from it we can draw some general conclusions as to the behavior of even
those circuits where it is difficult to isolate the two terminals across
which the negative impedance is developed. This chapter serves not
only as an extension of the previous techniques, but also as an introduc
tion to the difficult problems posed by nonlinear differential equations.
(b)
Fig. 151. (a) Basic negativeresistance oscillator; (6) typical voltampere character
istic with superimposed conductive load line.
151. Basic Circuit Considerations. The response of the voltage
controlled circuit of Fig. 15la, with which we are initially concerned, is
defined by the single node equation
dV
U
V dt + In =
(151)
where the bottom of the GLC circuit is taken as the reference node. Since
V = v a  V„ and /„ = /„ + in(t)
by differentiating Eq. (151) and collecting terms, the timevarying
response will be given by the solution of
d 2 v„ 1 ( r dv* din\ ,
d< 2 + C\ drrfij 1 "
LC
(152)
477
478 oscillations [Chap. 15
For smallsignal operation,
di„ _ din dvn _ dt)n
dt ~ dv„ dt ~ d dt
where g is the incremental conductance of the VNLR measured at the Q
point of the voltampere characteristic (point Q in Fig. 1516). With the
above substitution, the basic incremental defining equation may finally be
written as a quasilinear differential equation:
On + ^ (G + g)v n + L v n = (153)
The linearization of the differential equation, which results from the
implied assumption of a constant value of g, will not be valid once the
signal amplitude carries the operating path into the dissipative regions of
the voltampere characteristics. To find the amplitude of oscillation, the
nonlinear differential equation of the system must be solved. Before
treating this more difficult problem, we shall find the necessary and
sufficient conditions for the circuit of Fig. 151 to sustain almost sinus
oidal oscillations from the quasilinear equation.
Equation (153) has two roots, which are located at
Pw =~^(G + g)±j 4lCW^ G + g) * (154)
= a ± ja
These roots determine the nature of the timevarying response of the
negativeresistance oscillator, and therefore we shall investigate their
location as a function of g. In order to have a growing transient, a > 0,
which corresponds to
G < g (155)
Equation (155) can be satisfied only when the conductive load intersects
both the dissipative and the negative conductance portion of the charac
teristic. If there is only a single intersection on the negative portion of
the curve, the response will decay instead of increasing with time.
The second restriction which must be imposed on the roots, if the
response is to be oscillatory, is that u must be real. From Eq. (154),
AC
■£>(& + gY (156)
The boundaries delineating the inequality of Eq. (156) are two straight
lines, which satisfy
G +
&*>*
Sec. 151]
NEGATIVERESISTANCE OSCILLATOKS
479
These regions are shown in Fig. 152. We are interested in only that
portion of the plane below the g = —G line, because it is here that a self
excited signal is generated, and we are primarily concerned with the
narrow segment denning the sinusoidal oscillatory response.
Fig. 152. Modes of operation of the circuit of Fig. 151 as a function of the external
and internal conductances.
At the threshold of sustained oscillations, i.e., when G = —g, the
external conductance is tangent, at the Q point, to the voltampere char
acteristic and the negative conductance of the active element exactly
cancels the positive conductance of the tuned circuit. The purely
imaginary poles are located at
Pij = ±j ^m
For any other value of the external conductance lying in the oscillatory
zone, the load line also intersects the positiveresistance regions, and as
the oscillations build up, these segments will limit the amplitude to a
finite value.
When G is small compared with the magnitude of the negative con
ductance, u becomes imaginary and the circuit operates as an astable
multivibrator. In the relaxation region, the response is similar to that
discussed in Chap. 11, where the second energystorage element appeared
as part of the drivingpoint impedance of the active element instead of
as part of the external load. To return the operating point to the
sinusoidal oscillation region, we can load down the tuned circuit and
hence reduce its effective Q.
We might further note that as y/L/C is increased, the width of the
oscillatory region shrinks. A circuit operating properly under load
may switch into the relaxation zone once the load is removed. Since
480
OSCILLATIONS
[Chap. 15
the design of the tank controls the width of the oscillatory region, by
minimizing \/L/C, the performance of the oscillator becomes much less
critical.
The possible modes of system operation may also be explained by
following the migration of the system's poles as G is reduced toward
zero (as along the dashed line of
Fig. 152). Initially, the circuit is
dissipative and the two poles lie on
the real axis (Fig. 153). As G
decreases, the poles first coalesce
and then separate along the arcs
of a circle. This occurs at point
A in Figs. 152 and 153. When
G = —g, they lie on the imaginary
axis. A further reduction in G
permits an exponentially increasing
output, with the poles moving into
the right half plane; the negative
conductance now predominates. Eventually they again coalesce (point
C in Fig. 152) and separate along the positive real axis. Here the circuit
operates as a relaxation oscillator. Consider also the plot of the roots of
the negativeresistance switching circuit presented in Fig. 1126.
Example 161. To place some of the thoughts to be presented in this chapter in
their proper perspective, we shall now represent the tunedplate oscillator, which was
treated from the feedback viewpoint in Chap. 14, as a negativeresistance circuit.
Consider Fig. 154a when, in the active region, the grid winding is unloaded. With
the proper polarity connections of the feedback coil, the drive voltage becomes
M
e g = _ e p
Substituting into the controlled source of Fig. 154&, we see that the current flow is
proportional to the voltage developed across the two terminals. Hence the con
trolled source may be replaced by the negative conductance
Fig. 153. Pole migration along the
dashed path of Fig. 152 with decreasing
values of G. Points of interest corre
spond to those labeled in Fig. 152.
M
gm
leading to the equivalent model of Fig. 154c.
For sinusoidal oscillations, this negative conductance must lie within the appro
priate region of Fig. 152. By lumping r p together with the external conductance,
the restriction on the transconductance is given by
L + G + Jf > M^ > L + G
r p ' L L r p
Figure 154d illustrates how the negativeresistance region II is bounded by the dis
sipative segments; region I is where the grid conduction limits the amplification; and
Sec. 151]
NEGATIVERESISTANCE OSCILLATOES
481
region III is where the largesignal amplitude drives the tube into cutoff. Such
regions and their bounds are readily found from the piecewiselinear model.
T St 1  &m e g
(b)
(c) (d)
Fig. 154. (a) Tunedplate oscillator; (6) incrementalplate circuit model; (c) equiva
lent negativeresistance circuit; (d) voltampere characteristics as drawn through the
quiescent point of operation.
CNLR Oscillator. If a CNLR is employed as the energy source in a
negativeresistance oscillator, then the normal mode of operation is
satisfied by an external seriesresonant circuit (Fig. 155). Since this
is the dual of Fig. 151, the single defining
equation is written on a loop basis :
i£ + a/ +
H
I dt + Vn =
After differentiating and collecting terms,
we obtain
din
+
Fig. 155. CNLR oscillator con
figuration.
W ' L V du) dt ^ LC
(157)
which is the dual of Eq. (152). All arguments employed with respect to
the voltage solution of Eq. (152) are immediately applicable to the cur
rent solution of Eq. (157).
For example, the roots of the quasilinear differential equation, where
we define
— = r
din
482 oscillations [Chap. 15
are now located at
And the new boundary conditions are denned by the equations
For a > 0: R < r
For c real: R + J^ > r > R  J~
These should be compared with the conditions previously derived for the
VNLR circuit and with the plot of Fig. 152.
152. Firstorder Solution for Frequency and Amplitude. In order to
keep the nonlinearity in the forefront, Eq. (152), defining the complete
response of the basic voltagecontrolled negativeresistance oscillator of
Fig. 151, may be rewritten
S » + ?( 1+ gct)^ + zi W »= < 15 " 8 >
This is in the form of the Van der Pol nonlinear differential equation
v + ef(v)i> + coo 2 w =
For almost sinusoidal oscillations, ef(v) must remain small over the
range of interest. When it is zero, the Van der Pol equation reduces to
the equation of simple harmonic motion,
v + coo 2 v =
which has the solution
v = A cos (o>o< + 4>)
If the nonlinear term is large, then Eq. (158) can be easily solved only
for some specific analytic functions of tf(y).
To find the frequency and amplitude to a first approximation, we shall
apply the method of equivalent linearization. We shall assume that
over any single cycle the gross behavior of the system may be represented
by an equivalent linear circuit, and we shall neglect any variations taking
place during the individual cycle.
Choosing as a suitable trial solution for Eq. (158)
v„ = E(t) cos at
and substituting into Eq. (158) and collecting terms yields
[d*E „_ , E , G(, , ldi n \dE~[
Sec. 152] negativebesistance oscillators 483
For almost sinusoidal oscillations we may further assume that E(t)
remains essentially constant over any single cycle of the steadystate
response. Since
Eq. (159) reduces to
("• + eg) cos *  it i 1 + h £) 8in *  ° (15  10) 
To evaluate the average response, Eq. (1510) is now multiplied by
cos u>t and integrated over a complete cycle:
jT (" 2 + w) cos2 ui d ^  jT it( 1 + g a) sin at cos * d(  at)
=
If the circuit is adjusted close to the threshold of oscillation (to mini
mize the distortion of the sinusoid) , then the degree of limiting, and hence
the nonlinearity, is small. The coefficient of sin wt cos wt in the second
integral is this very nonlinearity. It plays a major role in determining
the amplitude, but since it is small, it will have a relatively minor effect
on the frequency. By assuming that this coefficient is essentially
constant over the period, the second definite integral would be identically
equal to zero. Consequently, the approximate frequency of oscillation,
given by the vanishing of the coefficient in the first integral, is the natural
frequency of the tuned circuit
1
0)0 =
Vlc
We shall now investigate E(f) by rewriting the fundamental circuit
equation. With the assumed cosinusoidal solution, Eq. (152) becomes
din \ n d 2 E , n dE,(\ 2 „\ _"
 df = [ C W +G Tt + (l  ^E^coswt
 UwC ^ + wGe\ sin wt (1511)
By again assuming that all coefficients are approximately constant over
the cycle, the first expression on the righthand side of Eq. (1511) may
be eliminated by multiplying through by sin wt and integrating over a
complete cycle. The equation of interest reduces to
w f 2 ' — sin wtd(wt) =  (2<oC^ + <&E\ P'sin 2 wtd(wt)
(1512)
484 Oscillations [Chap. 15
The lefthand side of Eq. (1512) may now be integrated by parts:
Jo d(o>i
A sin oit d{(d) = — u(i n sin ut)\ + co
>t) o
/•2x
/ in cos ait d(wt)
Jo
= 01 J i n cos (at d(oit) (1513)
Jo
But the righthand side of Eq. (1513) is proportional to the peak value
of the fundamental component of the current flow into the VNLR. Thus
the solution of Eq. (1512) is
omI\ m
Kt?
or
~~dt~2C (7lm + GE)
+ oiGE W
(1514)
(1515)
At equilibrium the oscillations are constant, dE/dt = 0, and
Ilm = —GE
The amplitude satisfying Eq. (1515) is found by plotting a describing
function, i.e., the fundamental component of the current into the VNLR
vs. the appliedvoltage excitation, and solving for the intersection
with the conductance line. Two typical curves appear in Fig. 156.
Fig. 156. Two describing functions for the basic nonlinear oscillator of Fig. 161:
curve 1 for a circuit biased in the center of the negativeconductance region, and curve
2 for a circuit biased in the positiveresistance region near the peak point.
Curve 1 appears when the circuit is normally biased within the negative
conductance region. For small signals, the current and voltage are out
of phase. As the amplitude of oscillation increases, the operating point
is driven into the dissipative regions for a portion of the cycle and the
fundamental current begins decreasing. The second curve corresponds
to a device normally biased within the dissipative region, such as at point
A or B in Fig. 151. For small signals, the drivingpoint impedance is
positive and the current is in phase with the applied excitation. A larger
amplitude, where the average conductance over a cycle is negative, per
mits sustained oscillations.
In order to see which of the multiple intersections represent stable
points of operation, we return to Eq. (1514) and consider the effects of
a small variation in E about the amplitude satisfying Eq. (1515).
Sec. 152] negativeresistance oscillators 485
w = ~"k [Ie + AI + G{E ' + AE)]
and I. and E. are the values at the equilibrium point. From Eq. (1514)
we know that 7, + GE, = 0. Consequently
^ m _ifM+flW (1516)
dt 2C\AE^ J
For stability, any change in E must establish the conditions that will force
a return to the equilibrium point. It follows that if the initial perturba
tion AE is positive and momentarily increases the output, dE/dt must
be negative in order for the voltage to decrease with time back to the
stable amplitude. This is possible only when the term in the parentheses
of Eq. (1516) is positive. The condition for stability becomes
f E + G>0 (1517)
Equation (1517) says that the total incremental conductance at the
stable point must be positive, or in other words, the circuit must appear
dissipative with respect to small variations about the equilibrium point.
In order for any oscillator to be selfstarting, the origin must be a
point of unstable equilibrium. The oscillator of curve 1 will build up to
point a, the single stable point. However, if the quiescent point is along
the positiveresistance segment, then the origin will be stable and the
circuit cannot start by itself (curve 2). Once the circuit is externally
excited to, or past, the unstable point b, the buildup will continue to the
second point of stable equilibrium, point /.
To evaluate the describing function analytically, the characteristic of
the VNLR may be approximated by a power series which is written with
respect to the quiescent point.
in = a,u„ + awn* + a s w„ 8 + • • • (1518)
where the various coefficients are found from the curve. Because of the
nature of the external circuit (which is a relatively highQ tank), only the
fundamental component of voltage may be developed across the terminals
of the nonlinear element. Furthermore, when the nonlinearity is small,
the fundamental is the only current component of interest.
Substituting v n = E x cos ut into Eq. (1518) and collecting the coeffi
cients of the cos utt terms results in
h = oj^! + HazES + HasES + ■ ■ ■ (1519)
For each value of the assumed voltage excitation, the current response
term is now known. Cross plotting gives the describing function shown
in Fig. 156.
486
OSCILLATIONS
[Chap. 15
The stable amplitude may also be found by realizing that at the thresh
old of oscillation, the negative conductance of the active element exactly
cancels the damping of the tuned circuit. Again, only the fundamental
components are of interest; at the harmonics the current and voltage are
in quadrature and do not represent power supplied to the tank. From
Eq. (1519), the average conductance of the VNLR over a cycle is
Gn = W 1 = <Xl + l a * El * + I atEii +
(1520)
Equation (1520) may be plotted as shown in Fig. 157 (the two curves
drawn correspond to the two currentdescribing functions of Fig. 156).
Fig. 157. Two possible conductance describing functions. Curve 1 holds when the
quiescent point is at the center of the negativeconductance region; curve 2, when the
quiescent point is in the positive region near the knee of the curve. These correspond
to the current describing functions of Fig. 156.
By superimposing the G line on the negative conductancedescribing
function, the amplitude satisfying G n + G = is finally found.
Example 162. A device having the almost symmetrical piecewiselinear character
istics of Fig. 158 is used in the basic oscillator circuit of Fig. 151. It is initially
biased in the center of the negativeresistance region, at Qi, where I c = 40 ma and
V„ = 20 volts.
By taking three coordinates on the curve, measured with respect to the Q point, we
can construct a cubic that will adequately approximate the characteristic. Choosing
the following points: ( — 10 volts, ma), (—5 volts, 20 ma), and (+5 volts, —20
ma), the approximating polynomial becomes
H = 5.33» + 0.0533«« ma
Thus the conductance describing function, as given by Eq. (1520), is
Oni = 5.33 + 0.04B,' millimhos
and it is plotted in Fig. 1586. When, for example, the conductance of the tank is
3.5 millimhos, the circuit will oscillate with a peak amplitude of 6.75 volts.
Suppose that the quiescent point is shifted to Qi, where V„ = 17 volts; by how much
would the amplitude change? Rather than solve for a new describing polynomial,
for the purposes of this discussion we can simply shift the previously derived curve to
correspond to the new origin. The shifted curve is denned by
ij 5.33(f  3) + 0.0533(»  3)» + A
Sec. 153]
NEGATIVERESISTANCE OSCILLATORS
487
where the constant A must be included to make i  when v = 0. This equation
reduces to
U « 3.89»  0.48»» + 0.0533»« ma
Consequently, the new conductive describing function is characterized by
Gni = 3.89 + 0.04JS, 2 millimhos
and the new peak amplitude stabilizes at 3.5 volts.
60
ra 40
" 20
.
\e 2
Qi
> 1
1
5 2
25 30 35 40 45 V
volts
(a)
2 4
i
i
10
1.0
* 2.0
G„ 3.0
Ex
G n y
4.0
^/&
iti
5.0
6.0
W
Fig. 158. (a) Voltampere characteristic; (6) the conductance describing functions for
Example 152.
153. Frequency of Oscillation to a Second Approximation. First
Method. In order to obtain a more accurate expression as to the fre
quency of oscillation, we shall now solve the Van der Pol equation with
the simple assumption that the voltage waveshape is periodic. This
equation is
v + ef(v)v + co 2 w =
where wo 2 = 1/LC and where
for the general circuit of Fig. 151. As the first step toward the evalua
488 oscillations [Chap. 15
tion of the oscillation frequency, the Van der Pol equation is multiplied
by the unknown solution and integrated over the unknown period.
f 2 ' wdut + e f 2 * f(v)vv dot + co 2 [ 2 *v i dust = (1521)
Let us now consider each term of Eq. (1521) individually, starting
with the one that contains the system's nonlinearity.
F(») = t J** f(v)vv dut (1522)
By expanding the voltampere characteristic into a power series,
* = OiW + ow 2 + a a v 3 + a 4 u 4 + • • •
the nonlinearity becomes
/(„) =(? + ^ = G + a! + 2a 2 w + 3a 3 t> 2 + • • ■ (1523)
The necessary assumption of a periodic solution leads to the expression
of v and ii as Fourier series.
v = V V k cos (host + 4> k ) (1524a)
and i, =  \ kaiVk sin (hat + 4>k) (15246)
Consequently vv consists solely of terms which are the product of two
timevarying signals in quadrature, a sinusoid and a cosinusoid; it does
not contain any dc terms. The substitution of v into Eq. (1523) and
the collection of terms will result in a dc component and a series of
cosinusoidal terms, i.e., the fundamental and all harmonics. The
integral of interest [Eq. (1522)] reduces to the evaluation of various
factors of the form
/ * (dc + cos nut) (cos koit) (sin moit) did
over a complete cycle of the fundamental where k, m, and n are nonzero
positive integers. But such an integral is always equal to zero. Thus
we have proved that
F( v ) = e [ 2r f(v)vvd<j>t =
Sec. 153] negativeresistance oscillators 489
Returning to Eq. (1521), the first integral may be evaluated by parts:
I vv doit = u I j—.v doit = oi I vd(v)
Jo Jo doit Jwto
\U — 2ir fat = 2r
oi v 2 dt (1525)
Jat0
\ut0
It follows directly from the argument employed with respect to the
integral of Eq. (1522) that
\ut = 2t
vv \ =
M0
and as a result
P* vv doit =  f*" t> 2 doit (1526)
The righthand integral may be identified as the meansquare value
of the derivative. Equation (1521) finally reduces to
f*' y 2 doit = oio 2 f 2 " v 1 doit (1527)
When the Fourier series of Eqs. (1524a) and (15246) are substituted
into Eq. (1527), only the squared terms would contribute to the final
answer: all cross products integrated over a cycle equal zero. The
evaluation of Eq. (1527), consistent with the assumed periodicity of the
solution, results in
H
V kWV k "
= Vim 1 ^ V * 2
t1
fc1
SO
y v k *
\"o/
»
y kw k *
 _£=i (1528)
Equation (1528) proves that when limiting introduces distortion
terms, the frequency of oscillation is always depressed below the nominal
resonant frequency of the tuned circuit, and that the greater the distor
tion, the greater the frequency depression. This equation is somewhat
difficult to use in the form derived, since for small amounts of distortion,
oi = wo. To simplify the expression, let us consider
«o 2
y w t(** div
k = l _ k^l
00 CO
^ k*V k * J k*V„*
1_4=1 Jt=l = »=1_ (1529)
*i *i
490 oscillations [Chap. 15
However,
1 _ w^ _ (too + o>)(o>o — to) ^ — 2 Ao>
O>0 2 O)o 2 O)o
where the approximation holds when o> is close to o>o and where the
frequency deviation is Au s u  o> . Furthermore, by restricting the
simplification of Eq. (1529) to signals containing only slight distortion,
the fundamental term greatly predominates. Because
Fi 2 » £ kW k *
k2
the denominator of Eq. (1529) may be approximated by Vi 1 .
The frequency deviation may finally be expressed in the form
Aw i£(fc 2 l)£I (1530)
O)
k2
For example, when the VNLR characteristics are symmetrical about the
Q point, only oddharmonic terms will be present in the output. If the
principal one is the third harmonic and if it is even 5 per cent of the
fundamental,
>xsx(^)'
^jX8X[^l^ 1.0%
Contributions of the higherharmonic components will depress the fre
quency even further.
Second Method. Our second method of finding the frequency depres
sion, because of the nonlinearity of the negative conductance character
istic, will be based on a power balance of the circuit. The ideal negative
resistance device, by its very nature, is unable to store any energy and is
also unable to supply any reactive power to the external circuit. Like
any characteristic that is a singlevalued function of the independent
variable, it does not exhibit hysteresis; consequently the line integral
of i dv over a cycle is
fi dv = (1531)
We shall therefore sum up the reactive power terms at each harmonic and
set the result equal to zero. The particular frequency at which this
condition is satisfied will be the frequency of oscillation.
As before, the unknown periodic voltage may be expressed as a Fourier
series. When it excites the tuned circuit of Fig. 15la, the reactive
component of the current at each harmonic will be given by
I' h = h sin fe = B k V k (1532)
Sec. 153]
NEGATTVEBESISTANCE OSCILLATORS
491
where B k is the susceptance at the fcth harmonic. I' k is in quadrature
with the voltage component of the same frequency. The trajectory
traversed on the voltampere charac
teristic, due to this current, is an
ellipse having the area
A' k = *V k I' k = *V k I k sin <f> k
This path is shown in Fig. 159.
During one period of the funda
mental, the harmonic circumscribes
this path A; times, tracing out the area
A k = TkV k I k sin <t> k = *hV k *B k
(1533)
But this is simply vk/2 times the reac
tive power. Since no energy is stored
by the VNLR, Eq. (1531) must be satisfied,
expressed as
V
Fig. 159. Negativeresistance char
acteristic showing the path traversed
by the harmonic components of volt
age and current.
This condition may now be
no »
V A t = = tVJB x + r ^ kV* 2 B k
tl *2
or, in terms of the reactive power,
Pis + y kP kB =
(1534)
(1535)
At some sufficiently high harmonic, usually at and above the second, the
tuned circuit appears capacitive. In order for Eq. (1534) to hold, JSi
must have a sign different from that associated with B 2 , B s , etc. This
is possible only if the tank appears inductive at the fundamental. We
have therefore proved once again that the frequency of oscillation must be
below the resonant frequency of the tuned circuit.
The susceptance of the tuned circuit may be expressed as
B = Cm
/ CO _ COo\
\oio u> )
(1536)
In the vicinity of resonance Eq. (1536) may be approximated by
Bi S 2C Ace
and at the harmonics Eq. (1536) reduces to
B k ^ Uo C
1
492 oscillations [Chap. 15
Substituting these two terms into Eq. (1534) and solving results in
£«_ _ I V (k*  1) ^
which is identical with Eq. (1530). Furthermore, since from Eq. (1534)
IiVi sin </>i = — ZklhVk sin <j> k
the depression in frequency is also given by
where <t>i is the phase angle of the fundamental current with respect
to the fundamental voltage.
Synchronization. The automatic balancing of the circuit, which shifts
the frequency to a point where the net reactive power is equal to zero,
may be utilized for frequency entrainment or synchronization. Suppose
that an external signal that is some rational ratio of the desired funda
mental is applied to the basic negativeresistance oscillator. Because
the tank is primarily reactive at this frequency, the additional power
injected (P,&) will be reactive and Eq. (1535) will become
Pi+ Y kP k + kP, k = (1538)
42
Depending on the phase of the injected signal, the effective reactive power
P,h can be either positive or negative; the frequency at which Eq. (1538)
is satisfied may be shifted in either direction.
For example, if P, k is inductive and large enough to cancel the other
harmonic terms, then the oscillation frequency could be returned to w .
On the other hand, if P, k is capacitive, the net harmonic power is
increased and the frequency would be further depressed. Essentially,
the injection of this additional power changes the effective susceptance
of the circuit.
154. Introduction to Topological Methods. Nonlinear systems which
are described by a differential equation of the second order may also be
studied with the air of various topological constructions. This form of
representation is important because it enables us to gain perspective as
to the totality of the possible behavior of the system without having to
solve the actual differential equation for the time response. The periodic
solution will appear as a closed contour on some graph. And as the
graphically presented nonlinearity will be used directly — without con
version to an approximating polynomial — grossly nonlinear as well as
Sec. 154]
NEGATIVERESISTANCE OSCILLATORS
493
quasilinear circuits may be treated. This is not to say that the topo
logical solutions do not have their own limitations; they do. In the
course of the following arguments some of the drawbacks will become
apparent.
As a convenient starting point, consider the secondorder linear differ
ential equation
x + bx + <ao 2 x =
(1539)
Phase plane
Two initial conditions, which give information as to the initial energy
in the system, are necessary for the complete solution; usually the
position and velocity at t = are specified.
For the simplest case of Eq.
(1539), we set 6 = 0, which re
duces the problem to one of simple
harmonic motion. The wellknown
solution for x in terms of time is
x = K cos (adt + <£) (1540)
But in order to evaluate the two
constants, we usually also find the
derivative.
Concentric
ellipses
Fig. 1510. Phase portrait of simple har
monic motion [Eq. (1539) with 6 = 0].
y = x = — u>oK sin (&>ot + 4>)
(1541)
Since time appears in Eqs. (1540) and (1541) merely as a parameter,
as an alternative form of presenting the answer, x may be plotted as a
function of x.
Equations (1540) and (1541) are parametric equations of an ellipse.
Eliminating t yields, as the solution of the original differential equation,
= 1
(1542)
The starting point of the plot on the x, x plane (hereafter called the phase
plane) is simply the coordinates given by the initial conditions. Since
two pieces of information uniquely determine any seconddegree curve,
one and only one ellipse, satisfying Eq. (1542), may be drawn through
each point on the plane. The totality of all such curves (Fig. 1510) is
called the phase portrait of the system.
Rotation of the trajectory about the origin is always in a clock
wise direction. Counterclockwise rotation would indicate a decreasing
displacement when the velocity is still positive and an increasing dis
placement for negative velocities — an obviously impossible situation.
Furthermore, these ellipses must cross the x axis at right angles. The x
494
OSCILLATIONS
[Chap. 15
intercepts are the points at which the velocity changes direction, and
consequently they must also be the points of the displacement maxima.
The above arguments lead directly to two additional conclusions:
1. All closed paths must encircle the origin.
2. All closed curves correspond to periodic motions.
The equilibrium point of the system is where x and x vanish. In
this example only the origin is stable, and it therefore is called a singular
point.
(a) (b)
Fig. 1511. Phase trajectories of the linear seconddegree equation, (a) Damped
oscillatory response; (b) the aperiodic solution.
If friction (or damping) is included in the system, then b > and
Eq. (1539) has two possible solutions.
1. The damped oscillatory wave:
x = Ke~" cos (coi< + <t>) (1543)
which exists when the roots are complex conjugates,
Pi. j = — a ± j"i
The phase portrait corresponding to Eq. (1543) and its derivative
appears in Fig. 151 la. As the oscillatory wave damps out, the trajec
tory describes a logarithmic spiral about the origin, circling it once
for each cycle of the cosinusoidal portion of the solution. Since the
spiral eventually terminates at the origin, this singularity is now called
a stable focal point.
2. The damped aperiodic wave which corresponds to the overdamped
case:
x = Kitr*' + K&*' (1544)
The damped aperiodic wave of Eq. (1544) cannot have more than one
Sec. 154] negativeresistance oscillators 495
point of zero velocity (excluding the trivial point at t = <»). Therefore
the phase trajectory will intercept the x axis once — it cannot encircle
the origin. In Fig. 15116, the origin becomes the stable nodal point
of the system.
Isocline Construction. The trajectories that represent the solution of
the differential equation on the phase plane may be plotted directly from
the equation by the method of isoclines. These are the locus of all points
in the phase plane where the curve has the same slope. By making the
substitution
Eq. (1539) may be rewritten
dy
dt
dx
y = di
by  o><?x (1545)
But in order to find the slope, Eq. (1545) will be divided through by y,
resulting in
„ dy dy dt , „x . fl .
S = Tx = diTx= h  m y < 1W6 >
Thus all points having the same slope S lie on a straight line through the
origin, satisfying the equation
Once the isoclines are drawn, shortline segments of the appropriate
slope may be marked off along each one, as shown in Fig. 1512. These
will serve as guides when drawing the phase portrait. Starting at any
initial point, a short section of the curve can be drawn having the slope
defined by the isocline passing through the starting point. From the
end of the first section, the next section is drawn, and so on, each inter
secting the next clockwise isocline with the proper slope until the curve
is completed. The totality of the individual segments is the actual
phase trajectory satisfying the initial conditions.
Such a construction is carried out in Fig. 1512 for the damped oscilla
tory case. For the purpose of illustration, we set co = 1 and b = y±,
reducing Eq. (1547) to
The initial conditions, which give the starting point of the single tra
jectory drawn, are x = — 3 and x = 5.
Returning to Eq. (1547), we might observe that the condition of an
496
OSCILLATIONS
[Chap. 15
infinite slope is satisfied along the x axis. This proves the contention
that the trajectories cross the x axis perpendicularly.
The most direct path to the stable point would be along an isocline.
Since the slope of the trajectory is identical with the slope of the isocline,
.
>i
c
n
II
1
o
■6
£/
^V^
%
%
ro
Ln
07
♦7
%
Hr""
s"
»/
>7
V
#
Lx?J_J5
2T 1
Initia
onditio
e \
7
af/
*>,
K
N$»
y£
\c
ns
c
sJH
<fr
\
V
1
^
^1
s?r
*v.
\
'
^
rO>
fej
^
'iK ,? '
so
\
fl ?60
N
5
—t
\
13
2
j
^
SO
__2
3
^
5
*
V
^
Jf
3
~
)
\
I
\
/
j
\
V
y
/
I
V
\
\
Fig. 1512. Isocline method of constructing a phase trajectory,
from Eq. (1547) we see that the necessary condition for such a path is
S =
S + b
But by multiplying out and collecting terms, there results the quadratic
S 2 + bS + o> 2 =
Except that it is a function of S, this quadratic is identical with the
characteristic equation from which the roots of the original differential
equation (1539) are found.
Straight lines having a slope equal to the roots are feasible only when
Sec. 154] negativebesistance oscillators 497
these roots are real, i.e., when the response is aperiodic. Two such
lines exist on each phase plane as shown in Fig. 15116. As they head
toward the origin, all trajectories approach the line corresponding to the
smallest root. It predominates, since it represents the long time decay
of the system. The larger root determines the initial rise, and its isocline
separates the trajectories that finally approach the origin in the second
quadrant from those that terminate in the fourth quadrant.
Unstable Equilibrium. If energy is supplied to the system at the proper
rate, for example, by connecting the negative drivingpoint resistance
(a)
Fig. 1513. Trajectories for negative damping.
(b) increasing aperiodic solution.
(b)
(a) Increasing oscillatory response;
across a GLC parallel circuit, the amplitude would increase rather than
decrease with time. In the region where the circuit may be approxi
mated by a quasilinear equation, i.e., before limiting, the trajectories
would diverge from the singularity at the origin, rotating clockwise as
they do so. The two cases of most interest are shown in Fig. 1513.
The increasing spiral represents the buildup of an almost sinusoidal
oscillator, and the aperiodic trajectory illustrates the switching path of a
multivibrator. If the system has some initial energy, the single tra
jectory of interest would be the one passing through the point on the
phase plane corresponding to the initial conditions. In the case of an
increasing exponential, the trajectories will approach the isocline cor
responding to the larger root of the secondorder equation, regardless
of how they diverge from the origin. If we are interested in the response
very far away from the point corresponding to the initial conditions,
then this particular isocline would closely approximate the path. Of
course, nonlinearities in the system may distort the trajectories long
before they ever reach this asymptote.
Period and Waveshape. Topological methods may also be employed
to convert the trajectory on the phase plane into the equivalent time
varying signal. We can construct a small line segment in the time domain
498
OSCILLATIONS
[Chap. 15
having the slope and position given by the corresponding point on the
phase plane. To the end of the first one, a second segment is added,
then a third, and so on. As we work our way along the trajectory, the
totality of all segments will give a rough approximation to the waveshapes
generated. This process is, of course, quite tedious, but necessary where
simpler methods cannot yield a satisfactory solution.
t '
X
1
2
t, sec
j
)
!
t
\
i
l
h
3
c
> '
/
/
/
/
1
5
Fia. 1514. Waveshape constructed from the trajectory of Fig. 1512.
Figure 1514 shows just such a construction for slightly more than one
complete encirclement of the origin by the trajectory of Fig. 1512. The
nitial point of the trajectory was chosen as the start of the construction.
Wherever the slope is large, large increments of position are permissible
over the associated line segment. Since this section of the curve does not
contribute much toward the period, relatively gross approximations
are justified. But when the operating point is moving very slowly, the
positional increments must be quite close together. To further reduce
the timing error, the average slope over each segment of the trajectory
was used in the construction of Fig. 1514.
The timing accuracy of this construction is very poor, and therefore it
should be used mainly to identify the waveshape. In the example shown,
the time for one complete cycle is almost 1 sec too long.
Sec. 155] negativeresistance oscillators 499
The time required for the operating point to pass between any two
points on the trajectory is found from the line integral:
<fdx = <f^dx = (f'dl=T b T a (1548)
In general, Eq. (1548) is evaluated by means of a graphical or a
numerical integration. For the complete period of any closed contour on
the phase plane, the integral becomes
T = <£dx (1549)
and it is taken over the complete trajectory.
156. Lienard Diagram. As the first step in finding the topological
solution of the Van der Pol nonlinear differential equation, we shall
normalize the basic equation (158) by dividing through by wo 2 ; this
leads to the result
Since the variables of interest are the voltage across the terminals of the
VNLR and its derivative, there should not be any ambiguity or confusion
of terms if we redefine v = dv/d(o>ot) and rewrite Eq. (1550) :
v + e'f(v)i> + v = (1551)
And following the procedure outlined in Sec. 154, the slope of the
trajectory, at each point in the phase plane, will be given by
«  £   !m i ±I WD
The isoclines defined by Eq. (1552) are no longer straight lines, but
are curves which depend on the nature of the nonlinearity. With
any given voltampere characteristic, f(v) may be evaluated for each
value of v. The isoclines are finally constructed, if necessary by plotting
individual points.
As this process is extremely tedious, we shall instead turn to the simpler
construction developed by A. Lienard. By means of a linear transforma
tion, from the phase plane to the Li6nard plane, the problem becomes one
of finding the normal to the trajectory and then simply striking an arc
to find a segment of the topographical portrait. The transformation
chosen is
z = i> + e'F(v) (1553)
where /(*) = ^ (1554)
500 oscillations [Chap. 15
By differentiating Eq. (1553) with respect to v and rearranging terms,
an alternative expression for the isoclines results:
„ di) dz ... ,
S = dv = dv ~ * m
Substituting S into Eq. (1552) leads to
dz _ _ v
dv v
From the linear transformation of Eq. (1553), the slope of the trajectory
Fig. 1515. Illustrating the means of evaluating F (v) .from the voltampere character
istics.
on the Lienard plane finally becomes
dz v
dv~ z  e'F(v)
and the slope of the normal to the trajectory is given by
z  e'F(v)
N =
dv
dz
(1555)
(1556)
Before we can construct the trajectories we must evaluate t'F(v).
For the basic circuit of Fig. 151,
dF(v)
= f(v) =G +
di
(1557)
dv Jy "' " ' dv
Integrating both sides with respect to v yields
F(w) = Gv + i = i  {Gv)
where v and i are measured from the Q point on the voltampere char
acteristics and where — Gv is the conductive load line passing through the
Q point. Equation (1557) says that F(v) is the difference in current
between the load line and the characteristic (as shown by the vectors in
Fig. 1515).
Sec. 155]
NEGATIVERESISTANCE OSCILLATORS
501
In replotting F(v), the ordinate would also be multiplied by «' to scale
the curve. This factor is dependent on the design of the tank and may,
in fact, be considered a defining parameter.
cooC
(1558)
The degree of the circuit nonlinearity is proportional to «'; the system
becomes quasilinear for very small values of «' and grossly nonlinear for
very large values. We conclude that for an almost sinusoidal oscillator,
.
zi+e'F(v)
Pi
Pa
P 3
A'FXv)
/ ^v r "
\
/ r3
/I3 V
r 2
\
r l
*1
Fig. 1616. Construction of the Lienard diagram.
L/C should be made as small as possible consistent with the other design
requirements. This supports the arguments made earlier with respect to
Figs. 152 and 153.
The final plot of t'F(v), which serves as the basis for the construction
of the Lienard diagram, appears in Fig. 1516. The procedure followed
in constructing first the normal and then the trajectory is based on
Eq. (1556). The various steps illustrated in Fig. 1516 are:
1. Plot the nonlinearity e'F(v), following the construction outlined
above, on the Lienard plane. This plot cannot be scaled in either
coordinate but must be drawn 1:1.
2. At any point on the plane pi, drop a perpendicular to e'F(v) at q\.
3. Draw a line from gi to the z axis at ri.
4. The vertical line segment
Piqi = 21  t'F{vi)
at that particular value of z x and V\. The horizontal line segment
qiri = vi
502
OSCILLATIONS
[Chap. 15
Hence, from Eq. (1556), the hypotenuse of the triangle ripiqi is the
normal to the trajectory passing through point p\.
5. Strike small arcs intersecting line segment piqi using point r t as
the center. These arcs are segments of the various trajectories on the
Lienard plane.
6. Starting at any initial point on the plane, strike an arc; from the
end of this arc strike another one; and then continue the construction
from one point to the next in a clockwise direction until the trajectories
converge into a closed contour.
A construction for four points appears in Fig. 1516, and complete
trajectories, cycling to the closed curve representing a periodic solution,
Fig. 1517. Lienardplane construction of the trajectory and the limit cycle and the
phaseplane limit cycle.
Sec. 155]
NEGATTVEBESISTANCE OSCILLATORS
503
appear in Figs. 1517 and 1518. In Fig. 1517 three different starting
points, two close to the origin and one at z = 30, v = 0, were chosen.
By following the procedure outlined above, three trajectories are con
structed, and these all eventually form the same closed contour encircling
the origin. This curve represents the only periodic solution of the sys
tem, and it is called a limit cycle. The path leading to the limit cycle,
,
2
s
.'"
""*>.
"N
,
V
/
1
\
1
I
"5B~
nstal
>le lin
I
lit cy
cle
*
,m»)
— 1
\
\
\
\
V
30
11*
,,'
*~"
"**<
\
T l
t
V
l
lr
Js
n ]
r /
0/2
0/3
4
6
70 .
sV.
S
t'y
V
fSt
able
imit (
ycle
Fio. 1618. Lienardplane construction illustrating the conditions yielding both stable
and unstable limit cycles.
from any starting point, is the transient portion of the solution. Two
trajectories in Fig. 1517 show the exponential buildup. One, which
starts with excessive amplitude, damps down to the final steadystate
solution.
The plot on the Lienard plane can be converted into a phase portrait
by performing the inverse transformation
v = z e'F(v)
which involves subtracting from each point on the Lienard trajectory the
value of t'F(v). This process is also carried out in Fig. 1517. Because
of the large degree of nonlinearity, it leads to the grossly distorted contour
shown.
The phase portrait, as well as the Lienard plot of an oscillatory system,
504 oscillations [Chap. 15
will exhibit one or more limit cycles. If all adjacent trajectories converge
on one contour (as in Fig. 1517), then it represents a stable oscillation.
Every unstable equilibrium condition is associated with some closed
contour from which the trajectories representing the transient response
diverge. It follows that stable and unstable limit cycles separate each
other. The origin may also be regarded as a solution which may be
either stable or unstable, depending on the nature of the trajectories
terminating on it. If the origin is unstable, the circuit is selfstarting.
If the origin is stable, the circuit must be triggered past the adjacent
unstable limit cycle.
Figure 1518 illustrates the nature of the unstable limit cycle, which
arises when the negativeresistance device is normally biased in its dis
sipative region near the knee of the voltampere characteristics. For
small signals, the circuit is stable and all trajectories cycle toward the
origin. However, after the signal becomes large enough so that the
circuit exhibits an average negative resistance, the buildup will proceed
toward the next stable limit cycle. These limit cycles correspond to the
fundamental amplitudes Et and E f in the describingfunction solutions
of Figs. 156 and 157.
The shape of the limit cycle, which depends on the size of «' for
any given voltampere characteristic and frequency, indicates the
nature of the time response. When e' is small, the limit cycle approaches
a circle or ellipse and the signal generated is almost sinusoidal (Fig.
1519o). When e' is large, the closed trajectory becomes almost rec
tangular and the time response is that of a relaxation oscillator (Fig.
15196).
Since the current through a capacitor is proportional to the derivative
of the applied voltage, the phase portrait is readily obtained experi
mentally. The VNLR voltage is applied to the x plates of a cathoderay
oscilloscope and a voltage proportional to the capacitor current to the
y deflection circuit.
At this point we might observe that the various topological construc
tions discussed in this chapter may also be applied to the solution of the
negativeresistance switching circuits of Chap. 11. At the low fre
quencies of operation, the external inductance in the VNLR switching
circuit predominates. This corresponds to a very large value of t'.
A trajectory similar to that shown in Fig. 15196 would be obtained.
Analogous response appears in the CNLR switching circuit. At the
higher frequencies, the effective internal energystorage element of the
device plays a more pronounced role, reducing d and introducing curva
ture into the trajectory. Thus the phase portrait explains the changes
in the behavior of the multivibrator with frequency and even its almost
sinusoidal time response when operating at the elevated frequencies.
Sec. 156] negativeresistance oscillators 505
Furthermore, the phase or Lienardplane construction can be used to
illustrate the transient switching path between the two states of a bistable
circuit. Since the stable points are the nodal points of the system, two
constructions would be necessary, one with each singular point as the
origin.
Wt
(a) (b)
Fig. 1519. Limit cycles and time response for two values of e'. (a) «' very small —
operation as an almost sinusoidal oscillator; (6) t large — operation as a relaxation
oscillator.
166. Summary. The complete solution of the nonlinear oscillator,
i.e., the waveshape, amplitude, and period, cannot be arrived at by
either the topological or the analytic method alone. These two tech
niques should not be regarded as independent, but rather as comple
mentary: where one is weak, the other is strong. By combining the
results, we are able to obtain almost any required information as to the
behavior of the system, information which cannot readily be obtained from
either method individually.
To find the frequency of oscillation, including the depression due to the
harmonic content of the output, we must turn to the analytic solution
where an explicit expression was derived [Eq. (1530)]. However, to
evaluate this equation, we must have some knowledge as to the wave
shape produced. If this is to be obtained analytically, then the non
linearity must be expressed as a power series. After the fundamental
amplitude is found from the describing function, the harmonic content is
evaluated by substituting back into this series. In simple cases, for
506 oscillations [Chap. 15
example, where the voltampere characteristic is almost symmetrical, the
accuracy requirements are satisfied by approximating the nonlinearity
by a cubic polynomial. The coefficients are relatively easy to evaluate.
However, in most cases a fifth or higherdegree equation is necessary
for an adequate description, and it becomes extremely tedious to solve
for the coefficients. Finally, we should note that the analytic method
does not give the waveshape of the output directly, but only as a Fourier
series.
On the other hand, the topological solution yields the peaktopeak
amplitude of the stable oscillation quite rapidly but supplies almost no
accurate information as to the frequency of operation. A good approxi
mation to the waveshape generated is easily constructed. From it one
could find the relative harmonic content of the output either by a graph
ical integration or by some schedule method.
Suppose that we assume that the period of the waveshape constructed
can be scaled to satisfy the frequency found from the analytic expression.
Then, from the topological construction, we obtain the waveshape and
the harmonic content and use this to find the corrected frequency. Thus
each mode of solution is employed where it best serves and, without
actually arriving at an analytic expression for the output voltage, we are
able to solve the problems posed.
PROBLEMS
151. (a) Prove that the vacuumtube Colpitts oscillator may also be treated as a
negativeresistance circuit.
(b) Convert the negative conductance appearing from plate to ground (found in
part a) to an equivalent negative conductance across the complete tuned circuit.
(c) State the bounding values of g m for which this circuit will produce an almost
sinusoidal output.
152. A transistor Hartley oscillator is to be converted into its negativeimpedance
equivalent circuit. Assume that the transistor has a very large basetocollector
transconductance and that the two coils are not coupled. Make all reasonable
approximations. Find the limits of r c /r' n for which the circuit will function as an
oscillator.
153. A unijunction transistor (Chap. 11) having 7 = 3, ni = 300, and m =
200 is biased in its active region and is used as the active element in the CNLR
oscillator of Fig. 155. The circuit is designed to operate at 50 kc with a tuned cir
cuit that has a loaded Q = 10. Specify the limiting values of L and C for which this
circuit will function as an oscillator.
154. (a) Specify the slopes and bounds of the regions of Fig. 154d if the tube has
n = 50, r p = 20 K, and r c = and if the conductance of the tank is 10~ 4 mho. The
plate supply is 100 volts, and the tube is biased in the center of its active region.
State all approximations made in the course of the analysis.
(b) Will the positive grid loading have any effect other than the reduction in gain?
Explain.
NEGATIVERESISTANCE OSCILLATORS
507
(c) What limitation must be imposed on the other tank parameters for proper cir
cuit operation if M = 0.2L?
166. Two negativeresistance characteristics have powerseries expansions about
the Q point given by
1. i. = 3.0»  0.2v» + O.OOIv* ma.
2. u = 3.0v + 0.2»» + O.OOlw 6 ma.
(a) Plot the current describing functions of these two curves on the same graph
and find the amplitude of oscillations if G = 2 X lO 4 mho. ; if G = 1 X lO" 8 mho.
(6) Repeat part a with respect to the conductance describing function.
166. This problem is designed to investigate the response of various negative
resistance oscillators from a consideration of the powerseries expansion. Assume
that the terms of interest are
»« = <*i» + a 2 f * + a 3 v' + a,v* + a,i» s
(a) Prove that in order for an oscillator to be selfstarting,
a, < G
(6) Prove that
a b >0
for any values of the other coefficients.
(c) Under what conditions will the describing function have a minimum at some
nonzero value of Ei? State the answer in terms of the relative size of the coefficients.
(<f) Under what conditions will the describing function have both a maximum and
minimum value of i for some nonzero value of Ei"!
167. Figure 1520 shows the piecewiselinear voltampere characteristic of a semi
conductor device. Find the approximating cubic polynomial holding when the quies
cent point is in the center of the negativeresistance region. Solve for the amplitude
of osculation when R = 400 and y/L/C  1,000.
Fig. 1520
168. Express the frequency deviation in terms of the harmonic current components
of the VNLR. Give the answer in a form analogous to Eq. (1530). State the rea
son for all approximations made in the course of the solution.
169. Consider a negativeresistance oscillator using a device described by the
polynomial
i = — 2.0w + 0.5»' ma
(a) Find the amplitude of the fundamental term when G = 1.0 millimho.
(b) Calculate the amplitude of the thirdharmonic current component with the
assumed sinusoidal excitation found in part a.
508 oscillations [Chap. 15
(c) When C = 100 it/A and when wo = 10 8 , evaluate the depression in frequency
due to the harmonic voltage developed.
(d) Repeat the above calculations for O — 1.8 millimhos.
1610. Use Eq. (1528) to calculate the approximate depression in frequency if
the distortion is great enough so that the terminal voltage is almost a square wave.
Even though the answer will be far from exact, it will still give a rough order of
magnitude.
1511. An oscillator with a nominal resonant frequency of at, = 10" and a signal
amplitude of 20 volts has a 5 per cent thirdharmonic and a 1 per cent fifthharmonic
component. The tuned circuit has a Q of 50 at an,
(a) By what percentage will the frequency be depressed?
(b) If C = 100 ii/xi, what is the phase angle of the fundamental?
(c) What is the amplitude of the injected seventhharmonic component that will
just return the frequency to o> ?
(d) The frequency is to be increased to 1.03oj by injecting a secondharmonic
signal. What is the necessary amplitude of the synchronizing signal? Calculate
the new phase angle at the fundamental.
1512. Find the phase trajectory of the following singleorder system under the
conditions given:
x + bx =
Plot all trajectories on the same graph and justify any jumps in x that appear.
(a) b = 5 and 2(0) = 1.
(b) b H and x(0~) V 2 , 2(0) = 3.
(c) 6 = +2 and x(0) = 10.
(<*) b = +H and 2(0") = 10, 2(0") 10.
Find the time response for parts b and d by means of a graphical construction from
the phase plane.
1513. Plot the phase portrait of the following equation on a normalized phase
plane; i.e., the coordinates are i/w and x.
x+ 2Aw<>x + «o ! 2 =
(a) Set A = 1.1.
(6) Set A = 3.
Pay particular attention to at least one path that falls between the straightline
trajectories.
1514. Repeat Prob. 1513 for
(a) A = 1.1.
(6) A = 3.
1615. The behavior of an undamped pendulum in the vicinity of its unstable
point is defined by
x — bx =
where b > 0. In this case the origin is called a saddle point.
(a) Prove that the trajectories are hyperbolas.
(b) Plot two pairs of trajectories when b = 4. Sketch the asymptotes and indicate
the direction of rotation of all curves.
1616. Assume that the plot drawn in Fig. 1517 corresponds to t' = 1. We now
wish to consider the response when t' =0.1 and when e' = 10.
(a) Construct the limit cycle on the Lienard plane for these two cases.
(b) Sketch the approximate waveshapes produced.
(c) By how much does the peaktopeak amplitude change as e' increases from 0. 1
to 1 to 10?
NEGATIVERESISTANCE OSCILLATORS
509
1517. Accurately construct, out of small line segments, the timevarying signal
represented by the limit cycle of Fig. 1517.
1518. Plot a Lienardplane trajectory and the equivalent phaseplane portrait for
the oscillator employing a device having the piecewiselinear voltampere character
istic of Fig. 1521. The quiescent point is located at V„ — 50 volts and I„ = 5 ma,
and the tank parameters are VX/C = 3,000 and R = 5,000 ohms. Sketch the
waveshape.
40 60
Fig. 1521
120 volts
1519. The load resistance of Prob. 1518 is changed to 10 K and the Q point to
Vo = 80 volts and I„ = 3 ma.
(o) Plot at least two Lienardplane trajectories showing the transient decay toward
the stable point.
(b) Is it possible to make this circuit oscillate? Give the expected peaktopeak
output if it is possible. If it is impossible, justify your answer.
1520. Solve Prob. 157 by the method of the Lienard plane.
BIBLIOGRAPHY
Andronow, A. A., and C. E. Chaikin: "Theory of Oscillations," Princeton University
Press, Princeton, N.J., 1949.
Belevitch, V.: "Theorie des circuits nonlineaires en regime alternatif," Gauthier
Villars, Paris, 1959.
Edson, W. A.: "Vacuumtube Oscillators," John Wiley & Sons, Inc., New York, 1953.
Kryloff, N., and N. Bogoliuboff: "Introduction to Nonlinear Mechanics," Princeton
University Press, Princeton, N.J., 1943.
le Corbeiller, P. : The Nonlinear Theory of the Maintenance of Oscillations, J. IEE
{London), vol. 79, pp. 361378, 1936.
Lienard, A.: Etude des oscillations entretenus, Rev. gin. ilec, vol. 23, pp. 901946,
1928.
Minorsky, N.: "Introduction to Nonlinear Mechanics," J. W. Edwards, Publisher,
Inc., Ann Arbor, Mich., 1947.
Oser, E. A., R. O. Enders, and R. P. Moore, Jr.: Transistor Oscillators, RCA Rev.,
vol. 13, pp. 369385, September, 1952.
van der Pol, B. : The Nonlinear Theory of Electric Oscillations, Proc. IRE, vol. 22,
no. 9, pp. 10511086, 1934.
NAME INDEX
Aaron, M. R., 384
Abraham, H., 288
Anderson, A. E., 356
Anderson, F. B., 476
Andronow, A. A., 509
Angelo, E. J., Jr., 66, 143
Bates, L. F., 427
Beale, I. E. A., 356
Belevitch, V., 509
Benjamin, R., 384
Bloch, E., 288
Bode, H. W., 476
Bogoliuboff, N., 509
Bollman, J. H., 476
Boothroyd, A. R., 221
Bothwell, F. E., 476
Brenner, E., 25
Briggs, B. H., 205
Brown, D. R., 95
Edson, W. A., 476, 509
Enders, R. O., 509
Esaki, L., 338, 356
Farley, B. G., 356
Feinberg, R., 288
Gibbons, J. F., 356
Ginzton, E. L., 476
Glegg, K., 314
Goodrich, H. C, 243
Guillemin, E. A., 25
Hall, R. N., 356
Herzo